interplay of spin-orbit coupling and electronic coulomb

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Interplay of Spin-Orbit Coupling and ElectronicCoulomb Interactions in Strontium Iridate Sr2IrO4

Cyril Martins

To cite this version:Cyril Martins. Interplay of Spin-Orbit Coupling and Electronic Coulomb Interactions in Strontium Iri-date Sr2IrO4. Strongly Correlated Electrons [cond-mat.str-el]. Ecole Polytechnique X, 2010. English.pastel-00591068

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Thèse présentée pour obtenir le grade de

DOCTEUR DE L’ECOLE POLYTECHNIQUE

Spécialité : Physique des Matériaux et Milieux Denses

Cyril MARTINS

Couplage Spin-Orbite

et Interaction de Coulomb

dans

l’Iridate de Strontium Sr2IrO4

Interplay of Spin-Orbit Coupling

and Electronic Coulomb Interactions

in Strontium Iridate Sr2IrO4

Soutenue publiquement le 26 Novembre 2010 à l’Ecole Polytechniquedevant le jury composé de :

Président du jury : Luca PERFETTI Ecole Polytechnique, PalaiseauRapporteurs : Alexander LICHTENSTEIN Hamburg Universität

Marcelo ROZENBERG Université Paris-Sud, OrsayDirectrice de thèse : Silke BIERMANN Ecole Polytechnique, Palaiseau

Cyril MARTINS

Couplage Spin-Orbite

et Interaction de Coulomb

dans

l’Iridate de Strontium Sr2IrO4

Interplay of Spin-Orbit Coupling

and Electronic Coulomb Interactions


2010

Centre de Physique Théorique (CPhT)Ecole Polytechnique

France

Remerciements

Il est assez paradoxal que les premières (et peut-être seules) lignes que tu liras, ô curieux et/ou courageuxlecteur, correspondent en fait pour moi, auteur, aux dernières pages que j’ajouterai dans ce recueil. Etc’est avec plaisir que je retrouve ma chère langue de Molière pour parachever ce qui a été une longue,et souvent difficile, entreprise.

En ce jour, l’heure est pour moi au bilan, je contemple avec une certaine satisfaction, mais aussiune petite pointe de nostalgie, le chemin parcouru pendant ces trois dernières années et que ces quelques200 pages ont pour vocation de résumer.

En ce jour, il y a bien sûr une grande fierté qui m’emplit quand je relis le titre “docteur de l’EcolePolytechnique” apposé sur la première page de cet ouvrage mais il y a aussi et surtout un souvenir, uneimage qui me revient en tête: celle d’une assemblée debout en amphi Carnot un vendredi après-midide novembre, le sourire aux lèvres, le regard soulagé et heureux, faisant résonner la salle de leursapplaudissements chaleureux. Cinq mois se sont peut-être écoulés depuis, mais je garde encore intacten ma mémoire et dans mon coeur le sentiment que j’ai ressenti à ce moment-là.

Et en ce jour, c’est à chacune de ces mains qui battaient l’air avec enthousiasme, mais aussi àcelles qui n’ont pu être là mais ont tout autant participé à ma réussite, que je voudrais témoigner mareconnaissance.

Je tiens à remercier tout d’abord Silke, ma directrice de thèse, pour m’avoir donné l’opportunitéde travailler dans un contexte scientifique de grande qualité pendant ces trois années. Merci d’avoircru en “ce jeune Supaéro qui avait abandonné les avions pour la mécanique quantique” dès le premierjour et merci surtout pour ton investissement et l’intérêt que tu as toujours porté à mon travail, mêmejusqu’à des heures incroyablement tardives. Merci aussi pour ton encadrement malgré tes fréquents etnombreux déplacements et pour ton enthousiasme communicatif, même dans les moments de doute oùmon pessimisme tendait à prendre le dessus.

Je tiens également à remercier mes rapporteurs Alexander Lichtenstein et Marcelo Rozenberg ainsique Luca Perfetti qui tenait le rôle de président du jury. Vous m’avez tous les trois fait un grand hon-neur en acceptant d’être présents à ma soutenance et en prenant le temps de lire attentivement monmanuscrit. Vos questions et remarques, aussi difficiles fussent-elles sur le moment, m’ont été autant deprécieux moyens pour améliorer mon travail et approfondir encore mes connaissances dans le domainede la physique des matériaux.

Ma reconnaissance va aussi à tous les membres du groupe Matière Condensée du Centre de PhysiqueThéorique (CPhT) de l’Ecole Polytechnique, que j’ai pu côtoyer depuis septembre 2007. Chacun d’entrevous a contribué à sa manière au résultat que vous voyez aujourd’hui.

En premier lieu, je souhaite remercier Antoine Georges, notre “chef de labo”, avec qui j’aurais aiméencore plus intéragir. Merci pour m’avoir accueilli au sein de cet équipe formidable, pour m’avoirdispensé tes cours à l’ENS, l’X ou au Collège de France et pour avoir toujours su me mettre à l’aiseau sein du laboratoire.

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Un grand merci aussi et surtout à tous mes collègues, avec qui j’ai partagé mes doutes, mes (fausses)joies, mes galères et mes découvertes au quotidien durant ces trois ans. Ce fut un véritable plaisirde travailler, collaborer, échanger mais aussi et surtout de rire et s’amuser avec vous. La grandeconvivialité que vous faites (ou avez fait) régner au labo a été un élément fondamental dans monépanouissement, au moins autant que votre enthousiasme et votre passion pour la physique que vousme transmettiez. Merci donc à Corinna Kollath, Veronica Vildosola, Donat Adams, Peter Barmettler,Pablo Cornaglia, Xiaoyu Deng, Michel Ferrero, Hartmut Hafermann, Igor Krivenko, Lorenzo de Leo,Luca de Medici, Olivier Parcollet, Dario Poletti, Alexander Poteryaev et Leonid Pourovskii.

Merci à mes prédécesseurs Jan et Lam pour leur exemple et leurs utiles conseils. Merci à messtagiaires Matthieu et Samuel. Merci à Loïg, mon compagnon de route (et successeur) avec qui j’ai“senti la ville” de Tokyo, découvert la face cachée de L.A. (Ah, Venice Beach by night !) et surtouttraversé le désert (de Californie. Il faudra qu’on retourne voir les champs de coton d’ailleurs !).

Enfin, un immense Merci au quatuor de post-docs qui m’ont énomément aidé et soutenu dans mestravaux et dans la rude tâche qu’a été la rédaction de ce manuscrit. Grazie Mille à Michele pour tonbel accent chantant qui me faisait voyager durant tes coups de téléphone interminables mais aussi etsurtout pour tes nombreux conseils et ton écoute. Vielen Dank à Markus pour tes “réponses à tout surtout et bien plus encore” et ton fameux “Pole Pole” qui m’a remonté le moral en septembre. Hvala àJernej pour ton optimisme débordant. Grâce à toi, j’ai appris que les problèmes pouvaient toujoursdevenir aussi simples qu’un “Ciao Ragazzi”. Merci à Jean-Sébastien pour ton accent Québecois qui mefera toujours rêver et pour l’efficacité de tes recherches sur Internet (surtout quand il ne s’agit pas dephysique !).

Mais le CPhT ne se résume pas qu’au groupe de Matière Condensée. Je tiens à remercier aussinos quatre secretaires (en particulier Fadila et Florence) et nos trois informaticiens (tout spécialementStéphane) sans qui le labo ne fonctionnerait pas aussi bien.

Je remercie aussi chaleureusement Christoph Kopper qui m’a permis de peaufiner mes connaissancesen physique mathématique et m’a fourni un soutien financier pour la fin de ma thèse.

Merci aux membres du groupe Hautes Energies pour “leurs discussions quotidiennes en anglais dansla salle café, si animées et si distrayantes”. Je remercie enfin Patrick Mora, le directeur du CPhT,et Marios Petropoulos du groupe des Cordes pour leurs signatures annuelles et leurs conseils lors desrenouvellements d’allocation de bourse de thèse à chaque automne.

A ce sujet, merci à l’Ecole Doctorale (EDX) pour son soutien financier pendant ces trois ans àtravers l’allocation internationale de thèse Gaspard Monge ainsi que pour l’allocation post-doctoral del’X qui m’a été atribuée et me permet de poursuivre actuellement mes recherches. Plus particulière-ment, merci à Fabrice et Audrey pour leur disponibilité et leur écoute.

Je souhaite enfin exprimer toute ma gratitude envers mes “sensei” Ferdi Aryasetiawan, TakashiMiyake, Rei Sakuma et Masatoshi Imada qui m’ont accueilli et fait découvrir leur culture lors de mesdeux séjours au Japon en 2005 et 2006. Plus encore, je leur suis très reconnaissant pour m’avoiraccueilli en stage post-doctoral depuis février et pour m’avoir aidé bien au delà de leurs attributionspendant ce séjour (tout particulièrement pendant les jours qui ont suivi le grand séisme de Sendaï).Collaborer avec vous est un grand plaisir et une source constante d’enrichissement, j’espère de toutcoeur pouvoir retourner au pays du Soleil Levant très bientôt pour poursuivre nos travaux ensemble “devive voix”.

Mes derniers mots iront à tous ces gens pour qui les pages qui suivent demeureront sûrement unroyaume quasi-incompréhensible mais qui ont contribué par leur soutien quotidien à cette réussite bienplus qu’ils ne peuvent l’imaginer.

Merci à Nicolas mon Binôme et Cyril mon alter ego. Merci à tous mes amis “râleurs” de Supaéro:Alexis l’improvisateur Toulousaing, Damien le financier devenu contrôleur aérien, les Vernonais Benoît

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et Julien, ce sacré Charles, Jérôme l’Alsacien, les deux Romain de Paris et Berlin. Merci à Sébastienl’astrophysicien expatrié et sa petite femme Charlotte (revenez nous vite des US !), à Pascal et Angelamon couple de “geeks” préféré. Mes pensées vont aussi à JC, notre grand montagnard, j’aurais aimépouvoir trinquer aussi avec toi à la fin de ma thèse.

Merci à mon Québécois Olivier qui, je crois, est le seul à comprendre mes divagations quantiques età mes compagnons de prépa qui sont aujourd’hui dans la même situation que moi il y a quelques mois :Olivier le Londonien qui m’a fait découvrir l’X sous d’autres aspects, Baptiste et Damien, les matheuxdont j’attends les soutenances avec impatience.

Merci à Cécile, Dominique Devernay, Christian Teichteil et Emmanuel Fromager, Pierre Pujol etDavid Dean pour m’avoir lancer dans cette grande aventure en me faisant partager votre passion aucours de ma scolarité.

Merci aux Xdoc et au Binet Impro de l’X qui m’ont accueilli les bras ouverts en 2007. Merci à latroupe OnOff pour sa joie, sa bonne humeur et son talent. Grâce à vous, j’ai pu continuer à cultiverle plaisir d’improviser sur scène et j’ai même pu brûler les planches à Paris... OnOff un jour, OnOfftoujours...

Je termine ces remerciements avec ces personnes qui ont souffert et vibré tout autant, voire mêmeplus, que moi pendant ces trois années. Ceux dont le soutien étaient là quoiqu’il arrive, et ceux dontles applaudissements résonnaient le plus fortement à mes oreilles en ce vendredi 26 novembre. A eux,je veux tirer mon chapeau et les mots me manquent pour vous dire à quel point vous compter pour moiet à quel point la saveur de la réussite n’aurait pas été la même sans vous. Merci à Papa, Maman, Jul,Val, Papys et Mamies ainsi qu’à tout le reste de la famille. Merci à vous tous et désolé si ces pagesdemeurent “une énigme” et ne vous aident pas plus à répondre à la fameuse question “Mais sur quoi tutravailles alors?” =)

Et non, je ne t’ai pas oubliée. Toi qui a dû me supporter tous les jours depuis septembre 2007 (avecparfois barbe et mauvaise humeur), qui m’a entouré de ton amour et de ton soutien à chaque instant,qui est venue me chercher à 3h00 du matin au labo en octobre et se demande encore si on écrit “planewaves” ou “planewaves”. Merci à ce cours d’anglais du 10 septembre 2007 qui nous a permis de serencontrer et merci à toi d’être là auprès de moi tout simplement. Spassiba Bolchoï ma Taniouchka.

le 26 avril 2011, à PalaiseauCyril M.

Introduction

Solid state physics is certainly the physical science which has the most impact on our daily life. Fromthe fabrication of the first transistor in 1947 to the discovery of giant magneto-resistance (GMR) in1988 [18, 22], the discoveries made in this field have led to numerous technological and industrial appli-cations. Current telecommunication and computing devices would not exist without the developmentof condensed matter physics. Even after all these years, solid state research still remains extremelyactive as many new applications will surely emerge from a better understanding of high-temperaturesuperconductors, spintronics devices and carbon nanotubes, to only name a few.

This fast-pace technological development would not have been possible without the joint effort ofinnovative experiments and good theories. Actually, theoretical condensed matter physics has keptbusy some of the greatest minds of our century. Indeed, understanding the large realm of phenomenadisplayed by solid state materials requires the development of new sophisticated theories. This is trueeven for the “simplest” of phenomenon. A good example of this false simplicity is given by insulatingmaterials. Experimentally, insulators are merely compounds which cannot conduct electricity, but arethey all insulating for the same reasons?

The first theory developed to explain insulating behavior was based on band theory. In thisframework, electrons are described as independent particles moving in an effective potential inducedby the crystal lattice and by their fellow electrons. The system can then be understood in termsof “energy bands” and an insulating state occurs only when all bands are filled. With this physicalpicture in mind, the development of “density functional theory” (DFT) [70, 93] in conjunction withthe local density approximation (LDA) has enabled physicists to improve our understanding of manycompounds.

However, the picture is not perfect yet. DFT calculations fail to capture the physics of sometransition metal oxides or rare-earth compounds, which contain open d and f -shells. In these “stronglycorrelated materials”, the previous single-particle picture is not suitable at all as the repulsive Coulombinteraction between electrons plays a significant role. In the extreme limit, the energy cost of electroniccorrelations may be so overwhelming that the system can be an insulator even if its energy bands arenot filled. Such a material for which electronic correlations prevent the motion of the electrons is calleda “Mott insulator ” [120, 121].

An important breakthrough in our understanding of this new class of insulators was made in theearly nineties with the development of “dynamical mean-field theory” (DMFT) [57, 98]. Within thisnew framework, the lattice model is mapped onto a local impurity problem embedded in an electronicbath. By combining DFT band structure calculations with DMFT in the so-called “LDA+DMFTformalism”, a successful description of strongly correlated materials was then possible.

However, physicists cannot just rest on their laurels as nature always provides new challenges,even for well-established theories. Strontium iridate (Sr2IrO4) is one of these challenging materialsthat evades our understanding. Whereas 5d-transition metal oxides are usually considered as “weakly”correlated, this material exhibits an insulating state despite its odd number of electrons per unit cell.This puzzling affair remained a mystery for decades. Recently, the model of “spin-orbit driven Mott

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insulator ” has been proposed [84]. According to this picture, the cooperative interaction betweenelectronic correlations and the strong spin-orbit coupling explains the insulating state of this material.

Sr2IrO4 is however not an isolated example. spin-orbit interaction was found to play a significantrole in the properties of a growing variety of correlated compounds such as strontium rhodate [107],iron-spinel [37] and many other iridium-based transition metal oxides [116, 125, 146]. In addition, therecent discovery of topological band insulators [21, 94, 118] has shown that spin-orbit coupling can alsomodify significantly the band structure of weakly correlated solids, thus leading to a distinct phaseof matter. Whereas spin-orbit interaction was commonly thought as a “small relativistic correction”in solid state physics and even more so for strongly correlated materials, it appears now that thisassumption has been a mistake.

This thesis takes all this importance in this new context where studying the interplay betweenthe electronic correlations and the spin-orbit interaction has become essential. The main purposeof our work was to study the paramagnetic insulating phase of Sr2IrO4 within LDA+DMFT. Thisstudy required the extension of the current implementation of LDA+DMFT to take into account thespin-orbit coupling. In regards of this objective, this thesis is organized in two parts:

• In the first part, we introduce definitions and concepts that are of great use to the study ofstrongly correlated materials. More precisely, the first chapter is focused on density functionaltheory (DFT) and dynamical mean-field theory (DMFT). Having presented these two theoriesindependently, we then explain in the second chapter how to treat correlations in materialswithin the LDA+DMFT formalism. The implementation of LDA+DMFT developed in thelinearized augmented planewaves (LAPW) framework by Aichhorn et al. [1] is then described.This description is essential since our work particularly consisted in extending it so that thespin-orbit interaction may be included (more precisely, to define the Wannier orbitals on whichthe local impurity problem is based). A general presentation of this new “LDA+SO+DMFTimplementation” is finally given in the third chapter after a brief review of the effects of thespin-orbit coupling in atoms and solids.

• The second part is devoted to the 5d transition metal oxide Sr2IrO4 whose study has motivatedthe technical developments achieved in this thesis. In the fourth chapter, we give an exhaustivereview of the existing experimental and theoretical works performed on this compound, in orderto put our results into context. Our LDA+DMFT study is then presented in the last chapter. Onthe one hand, we confirm that Sr2IrO4 is a Mott insulator in its paramagnetic phase. On the otherhand, we highlight the respective roles played by the spin-orbit interaction and the structuraldistortions to reach the Mott insulating state. By systematically varying the correlation strengthin the absence and presence of both these elements, we indeed argue that only their actingtogether may open the Mott gap in Sr2IrO4.

Finally, this thesis ends with a series of appendices, where additional information on crystal fieldeffects in the presence of spin-orbit coupling, technical issues about the new implementation of projectorscheme and a discussion on many-body treatment of the spin-orbit interaction, invoking spin-same-orbitand spin-other-orbit terms, can be found.

Contents

Acknoledgements - Remerciements iii

Introduction vii

Table of contents xii

I Methods 1

1 A (not so) brief introduction to the domain 31.1 Basics of solid state physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Density Functional Theory (DFT) . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 The Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.3 The exchange-correlation energy and the Local Density Approximation (LDA) . 81.2.4 Success and limitations of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Electronic correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.1 Introduction to strong correlations and Mott insulators . . . . . . . . . . . . . . 101.3.2 The Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.3 Examples of strongly correlated materials . . . . . . . . . . . . . . . . . . . . . 13

1.4 The Dynamical Mean-Field Theory (DMFT) . . . . . . . . . . . . . . . . . . . . . . . 141.4.1 Introduction to the theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.2 Limits in which DMFT becomes exact . . . . . . . . . . . . . . . . . . . . . . . 171.4.3 Impurity solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.4 Theory of the Mott transition within DMFT . . . . . . . . . . . . . . . . . . . 19

2 Combining DFT-LDA calculations with DMFT: the LDA+DMFT approach 232.1 The LDA+DMFT formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.1 General description of the method . . . . . . . . . . . . . . . . . . . . . . . . . 232.1.2 The double-counting correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1.3 Choice of the localized basis set . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1.4 Approximations in the LDA+DMFT method . . . . . . . . . . . . . . . . . . . 29

2.2 Introduction to (linearized) augmented planewaves ( (L)APW ) . . . . . . . . . . . . . 302.2.1 Wien2k, an all-electron full-potential LAPW method . . . . . . . . . . . . . . . 302.2.2 APW and LAPW bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.3 The LAPW basis with local orbitals or (L)APW+lo basis in Wien2k . . . . . . 35

2.3 Projection onto Wannier orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.1 Wannier functions: definition and calculations . . . . . . . . . . . . . . . . . . . 362.3.2 Projectors on Wannier functions within the (L)APW+lo basis of Wien2k . . . . 38

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3 Taking into account the spin-orbit interaction in LDA+DMFT 41

3.1 Basics on the spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1.1 Derivation of the spin-orbit coupling term . . . . . . . . . . . . . . . . . . . . . 413.1.2 Effects on atomic orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1.3 Atomic d orbitals in cubic symmetry and spin-orbit coupling . . . . . . . . . . 45

3.2 Effects of the spin-orbit coupling in solids . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.1 Dresselhaus and Rashba terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.2 New domains involving the spin-orbit coupling . . . . . . . . . . . . . . . . . . 49

3.3 Implementation of the spin-orbit coupling (SO) in LDA+DMFT . . . . . . . . . . . . . 503.3.1 How the spin-orbit interaction is included in Wien2k . . . . . . . . . . . . . . . 503.3.2 Consequences on the definition of the Wannier projectors and on DMFT equa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4 Summary: Our “LDA+SO+DMFT” implementation within the LAPW framework . . 54

II The paramagnetic insulating phase of Strontium Iridate 59

4 Short review on strontium iridate (Sr2IrO4) 61

4.1 Crystal structure of Sr2IrO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Experimental evidence for an insulating state . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.1 Transport measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2.2 Optical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.3 Spectroscopy measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.4 Heat properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Theoretical models for the insulating state . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.1 The “spin-orbit driven Mott insulating” model . . . . . . . . . . . . . . . . . . . 67

4.4 Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.4.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.4.2 Model for the canted-antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . 69

4.5 Still open questions about Sr2IrO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.5.1 Specificities of the electrical transport in single crystal . . . . . . . . . . . . . . 694.5.2 Temperature dependence of the optical gap . . . . . . . . . . . . . . . . . . . . 70

5 The Sr2IrO4 Mott insulator 71

5.1 Effects of the spin-orbit coupling & the distortions within DFT-LDA . . . . . . . . . . 725.1.1 Case 1: “Undistorted ” Sr2IrO4 without spin-orbit coupling . . . . . . . . . . . 725.1.2 Case 2: Modifications in the Kohn-Sham band structure induced by the spin-

orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1.3 Case 3: Modifications in the electronic band structure induced by the distortions 885.1.4 Influence of the spin-orbit coupling and the distortions together . . . . . . . . 94

5.2 Effects of the spin-orbit coupling & the distortions within LDA+DMFT . . . . . . . . 1015.2.1 Case 1: “Undistorted ” Sr2IrO4 without spin-orbit coupling . . . . . . . . . . . . 1025.2.2 Case 2: Distortions leading to orbital polarization . . . . . . . . . . . . . . . . 1065.2.3 Case 3: Spin-orbit coupling and the reduction of the effective degeneracy . . . 1105.2.4 The Mott insulating state in Sr2IrO4 . . . . . . . . . . . . . . . . . . . . . . . . 115

Conclusion 119

CONTENTS xi

III Appendices 121

A Atomic d orbitals and spin-orbit coupling: Complements 123A.1 Atomic d states of a metal in an octahedral ligand field . . . . . . . . . . . . . . . . . . 123

A.1.1 Case of an elongated or compressed octahedron . . . . . . . . . . . . . . . . . . 124A.2 Cubic symmetry and spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 126

A.2.1 Effect of the spin-orbit interaction beyond the TP-equivalence approximation . 126A.2.2 Effect of a tetragonal splitting within the TP-equivalence approximation . . . . 128

A.3 Evaluation of the spin-orbit coupling constant and the tetragonal splitting in “undis-torted ” Sr2IrO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

B The self-energy of Sr2IrO4 131B.1 Relation between the spectral function and the self-energy . . . . . . . . . . . . . . . . 131B.2 The self energy of a 3/4-filled two-band Hubbard model in the atomic limit . . . . . . 132

C Structure and conventions in Wien2k 135C.1 General structure of the Wien2k package . . . . . . . . . . . . . . . . . . . . . . . . . . 135C.2 Conventions for the symmetry operations in Wien2k . . . . . . . . . . . . . . . . . . . 136

C.2.1 Symmetry operations T and local rotations Rloc . . . . . . . . . . . . . . . . . 136C.2.2 Representation of the symmetry operations by Euler angles . . . . . . . . . . . 137C.2.3 Standard computation of the rotation matrices . . . . . . . . . . . . . . . . . . 138C.2.4 Computation of the rotation matrices in Wien2k . . . . . . . . . . . . . . . . . . 140

C.3 Conventions for the coefficients Aναlm(k, σ), Bναlm(k, σ) and Cναlm (k, σ) . . . . . . . . . . . 141

D Description of dmftproj 143D.1 General structure of dmftproj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143D.2 Description of the master input file case.indmftpr . . . . . . . . . . . . . . . . . . . . . 144D.3 Execution of the program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

D.3.1 In order to perform a DMFT calculation . . . . . . . . . . . . . . . . . . . . . . 145D.3.2 In order to calculate a momentum-resolved spectral function . . . . . . . . . . . 146

D.4 Description of the output files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146D.4.1 Output to perform a DMFT calculation . . . . . . . . . . . . . . . . . . . . . . 146D.4.2 Output for calculating a momentum-resolved spectral function . . . . . . . . . 146

E Symmetry operations and projectors 147E.1 Space groups and time reversal operator . . . . . . . . . . . . . . . . . . . . . . . . . . 147E.2 Properties of an antilinear operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148E.3 Action of a space group operation on the projectors . . . . . . . . . . . . . . . . . . . . 150

E.3.1 Action of a space group operation on the Bloch states . . . . . . . . . . . . . . 150E.3.2 Transformation of the projectors under a space group operation . . . . . . . . . 151

E.4 Local quantities and sum over the irreducible Brillouin zone: . . . . . . . . . . . . . . . 154E.4.1 Case of a paramagnetic compound . . . . . . . . . . . . . . . . . . . . . . . . . 154E.4.2 Case of a spin-polarized calculation . . . . . . . . . . . . . . . . . . . . . . . . . 157

F From many-body spin-orbit interactions to one-electron spin-orbit coupling 161F.1 The spin-same-orbit and spin-other-orbit interactions . . . . . . . . . . . . . . . . . . . 161

F.1.1 Introduction of these interaction terms . . . . . . . . . . . . . . . . . . . . . . . 161F.1.2 Pauli matrices representation of a general interaction . . . . . . . . . . . . . . . 163

F.2 Spin-Hedin equations in their most general form . . . . . . . . . . . . . . . . . . . . . . 164F.2.1 A brief history on Hedin’s equations . . . . . . . . . . . . . . . . . . . . . . . . 164F.2.2 Spin-Hedin’s equations for a general interaction term . . . . . . . . . . . . . . . 165

xii CONTENTS

F.3 Generalized Hartree potential for our system . . . . . . . . . . . . . . . . . . . . . . . . 167F.3.1 The Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167F.3.2 The spin-same-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 168F.3.3 The spin-other-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

F.4 Generalized Fock terms in our system . . . . . . . . . . . . . . . . . . . . . . . . . . . 170F.4.1 The Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170F.4.2 The spin-same-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 171F.4.3 The spin-other-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

F.5 Expression of the screened interaction W . . . . . . . . . . . . . . . . . . . . . . . . . . 172F.5.1 Matrix approach & Polarization computation . . . . . . . . . . . . . . . . . . . 172F.5.2 Computation of the screened interaction W . . . . . . . . . . . . . . . . . . . . 173

Bibliography 175

Part I

Methods

1

Chapter 1

A (not so) brief introduction to thedomain

What is a “strongly correlated material ”? The best way to answer this question would certainly be togive an exhaustive list of the compounds which are considered so and the solved and still open questionsrelated to them. With such an approach, we could cite the transition metal and transition metal oxides,the cuprates or new iron-based superconductors, the rare-earth and actinide or lanthanide compounds.However, making such an exhaustive overview is a really tough task, above all in a few pages, and forthe interested reader we particularly recommend the review of Imada et al. [76]. Nevertheless, despitethe large number of compounds and the variety of their features, some general behaviors can still beextracted and finding the key-physical concepts which are hidden below this diversity of phenomena isprecisely the aim of the theoretical branch of this scientific domain.

In this chapter, we introduce the main physical ideas which are at stake in this field. Starting from ashort reminder of elementary notions which are of great use in all the condensed matter physics, we thenpresent the “density functional theory” (DFT) which was an important breakthrough in the domainin the early sixties. To understand the limits of this theory, the concept of “electronic correlations”must be introduced and is the subject of the second part of this chapter. The Hubbard model is thenpresented and the Mott physics is described. The last part of this chapter is devoted to the “dynamicalmean-field theory” (DMFT) which is now well-established in the community to describe the correlatedstates in materials.

1.1 Basics of solid state physics

Any solid is a polyatomic system which is composed of two coupled subsystems: a set of Nn atomicnuclei and a set of Ne electrons, these two entities being of the order of a few Avogadro’s number NA =6.022× 1023. The quantum mechanical description of this system relies on the following Hamiltonian:

H =

Nn∑

i=1

− ~2

2Mi∇2

Ri+

Nn∑

i,j=1i<j

(Zie)(Zje)

4πε0|Ri −Rj |

+

Ne∑

j=1

− ~2

2m0∇2

rj+

Ne∑

i,j=1i<j

e2

4πε0|ri − rj |−

Nn∑

i=1

Ne∑

j=1

Zie2

4πε0|Ri − rj |

(1.1)

where Mi is the mass of the ith nucleus, Zie its charge and m0 the electronic mass. The first two termscorrespond to the kinetic energy of the atomic nuclei and the Coulomb interaction energy between

3

4 CHAPTER 1. A (NOT SO) BRIEF INTRODUCTION TO THE DOMAIN

them. The third and fourth terms are the same physical quantities for the electrons and the fifth termstands for the Coulomb interaction energy between the electrons and the nuclei.

However, the mass of the nuclei is several orders of magnitude larger than that of the electrons:m0/Mi ≈ 10−4-10−6. As a result, a solid is described by two extremely different dynamical regimes,the velocities of the nuclei being considerably smaller than those of the electrons. The “adiabatic” or“Born-Oppenheimer approximation” takes advantage of this physical observation and allows studyingthe dynamics of the electrons separately from that of the nuclei. The initial problem (1.1) reduces thento a system of Ne interacting electrons in the static electric field V (r) induced by the fixed Nn nuclei:

H =

Ne∑

j=1

[− ~2

2m0∇2

rj+ V (rj)

]+

e2

4πε0

Ne∑

i,j=1i<j

1

|ri − rj |. (1.2)

In the following, we will consider only crystalline solids, in which the atomic nuclei are arrangedin an orderly periodic pattern in the three spatial dimensions. We furthermore assume that crystalsare infinite in space. As a consequence, it is possible to find a set of discrete translations and a set ofsymmetry operations – like rotations and reflections – which leave the system invariant after applyingany of these operations to it. The former defines the “Bravais lattice” B of the compound and thelatter its “crystallographic point group” [2, 16, 25].

By performing all these approximations, we have neglected the dynamic lattice deviations, the im-pact of disorder or defaults and the surface effects, which are important subjects in solid-state physicson their own. Nevertheless, even such an idealistic description of the problem remains almost impos-sible to solve because of its many-body character. Each of the 1023 electrons indeed interacts with allthe others via the Coulomb interaction – the last term of (1.2) – : their behaviors are “correlated ” witheach other.

The simplest way to take into account this effect is to approximate the last term of (1.2) in a“mean-field approach”. This is precisely the aim of the “Hartree approximation”, in which each electronfeels only the electric field induced by the mean charge density ρ(r) corresponding to all the otherelectrons:

H ≈Ne∑

j=1

[− ~2

2m0∇2

rj+ V (rj) + VH(rj)

]with VH(r) =

e2

4πε0

∫

R3

d3r′ρ(r′)

|r− r′| . (1.3)

Consequently, the Hamiltonian becomes separable and it is then possible to solve the problem by usingstandard methods developed for independent particle system.

Within this independent particle picture, the periodicity of the lattice results into two main conceptswhich form the base of solid-state physics:

Bloch’s theorem (1929) [24] which states that the wavefunction of a particle or pseudo-particle canbe written as:

ψk(r) = eik.ruk(r) with uk(r + R) = uk(r) ∀ R ∈ B (1.4)

where k is a reciprocal lattice vector which belongs to the “first Brillouin zone” (1BZ) of thecrystal and is called the “crystal momentum” of the particle. Bloch functions can also be writtenin the following form, where the sum runs over all the reciprocal lattice vectors K:

ψk(r) =∑

K

ck-Kei(k-K).r. (1.5)

1.2. THE DENSITY FUNCTIONAL THEORY (DFT) 5

The formation of “band structure”: In the crystal, electrons are confined to some intervals of en-ergy, and forbidden from other regions. According to Pauli’s principle, each “band ” can accom-modate two electrons with opposite spins by unit cell. Two different kinds of solid can then bedistinguished:

• the “metals”, whose band structure contains one or several partially-filled bands. An itin-erant behavior of the charge carriers is then realized and the energy εF up to which thebands are filled is called the “Fermi level ”. The density of states at the Fermi level of sucha compound is finite.

• the “band-insulators”, with an even number of electrons by unit cell, for which all the en-ergy bands are filled. These compounds are characterized by an “energy gap”, which is theenergy difference between the top of the highest occupied band – or “valence band ” – andthe bottom of the lowest unoccupied band – or “conduction band ” –. The density of statesinside the gap is of course zero.

Before concluding this section, we would like to focus the reader’s attention on the “mean-field ”concept, introduced to establish equation (1.3). To treat the many-body Hamiltonian (1.2) of theelectronic system, we have considered an equivalent problem of independent particles, described by aset of single-particle Schrödinger-like equations:

Hψk(r) = εnkψk(r) with H =

Ne∑

j=1

[− ~2

2m0∇2

rj+ V (rj) + VH(rj)

](1.6)

in which the two-body interaction is replaced by the effective Hartree potential VH(r) based on thefollowing approximation:

e2

4πε0

Ne∑

i,j=1i<j

1

|ri − rj |≈

Ne∑

i=1

VH(ri) =

Ne∑

i=1

e2

4πε0

∫

R3

d3r′ρ(r′)

|ri − r′| . (1.7)

This describes the effect of a mean-field interaction which becomes a functional of the electronic densityρ(r) defined as:

ρ(r) =∑

εnk≤εF

|ψk(r)|2. (1.8)

Since the density of the electrons ρ(r) is precisely unknown when one intends to solve equations(1.6), the problem must be solved iteratively by performing a self-consistency loop: starting from aninitial guess of the mean density, the eigenstates ψk(r) of (1.6) are found and the charge density asso-ciated to the corresponding ground-state is calculated with (1.8). The Hamiltonian (1.6) is then solvedagain with this new guess of ρ(r). The iterations stop when the convergence is achieved.

However, going beyond the mean-field approximation is essential in electronic structure calculationsto get the quantitative physics out of the solution. From this point of view, the real breakthroughin the field is represented by the “density functional theory” (DFT), which is the subject of the nextsection.

1.2 The Density Functional Theory (DFT)

The formalism of the “density-functional theory” (DFT) was introduced by Hohenberg and Kohn in1964 [70]. In 1965, Kohn and Sham established a set of equations to treat any interacting electron


system within this formalism [93]. Nevertheless, as we will see, some drastic simplifications in themany-body problem must be introduced to use in practice this approach. Despite these approxima-tions, this scheme has turned out to be reliable for describing the ground-state of numerous compoundsand has become the most popular ab initio method in quantum chemistry and solid-state physics tostudy electronic structures.

The brief overview of DFT presented here follows the traditional – historical – approach. A thoroughintroduction to this theory can be found in [32, 80]. We also mention, for the interested reader, thealternative presentation of [14], based on the analogy with thermodynamics.

1.2.1 The Hohenberg-Kohn theorems

In 1964, Hohenberg and Kohn formulated two theorems [70], which formally justified the use of thedensity ρ(r) as the basic variable in determining the total energy of a system composed ofNe interactingelectrons and described by the Hamiltonian (1.2). Such a statement was a real breakthrough inthe domain, since the ground-state of such a system is typically described by 3Ne variables in thewavefunction:

H|Ψ〉 = E|Ψ〉 with |Ψ〉 =∫

Ψ(r1, . . . , rN )|r1, . . . , rNe〉d3r1 . . . d3rNe (1.9)

whereas the electronic density distribution counts only 3 parameters:

ρ(r) = Ne

∫|Ψ(r, r2, . . . , rNe)|2d3r2 . . . d3rNe . (1.10)

First Hohenberg-Kohn theorem

Using the index “0” for ground-state quantities, the first Hohenberg-Kohn theorem states:

The external potential V (r) is (to within a constant) a unique functional of ρ0(r); sincein turn V (r) fixes the Hamiltonian H, we see that the full many particle ground-state is aunique functional of ρ0(r).

Thus it exists a one-to-one mapping between a given external potential V (r) and the ground-state density ρ0(r). This theorem also implies that all the ground-state properties of the interactingparticle system are exactly determined through the knowledge of ρ0(r). We will not display here thehistorical proof of the theorem, which shows the existence of this relation by reductio ad absurdum.No explicit formula to calculate the potential V (r) from ρ0(r) is actually known and in practice, someapproximation must thus be used.

Second Hohenberg-Kohn Theorem

The second Hohenberg-Kohn theorem extends the Rayleigh-Ritz variational principle for all densitydistributions ρ(r) which can be represented by Ne-body wave-function. The ground-state energyE0 = E[ρ0(r)] indeed minimizes the following functional of the density:

E[ρ(r)] = F [ρ(r)] +

∫V (r)ρ(r)d3r (1.11)

where F [ρ(r)] is a universal functional which contains the potential energy of the electron-electroninteractions and the kinetic energy of the interacting electrons.1 Since the expression of the Hartree

1This functional is said “universal” since it is identical for any system of Ne interacting electrons. In the formulationof Levy and Lieb [102, 106], F [ρ(r)] can be formally expressed as the minimum of the expectation values of the exact


energy as a function of the density is explicitly known, the functional F [ρ(r)] can be further decomposedinto:

F [ρ(r)] =1

2

∫d3r

∫d3r′

e2

4πε0

ρ(r)ρ(r′)

|r− r′| +G[ρ(r)]. (1.13)

Even with this last expression of F [ρ(r)], the variational principle still remains tricky to solve, sincean explicit expression of G[ρ(r)] in function of ρ(r) remains unknown.

1.2.2 The Kohn-Sham equations

In 1965, Kohn and Sham suggested to solve the complex interacting electronic system by using an“effective model of independent particles with the same ground-state density” [93]. In the same spiritas previously explained about the Hartree – and Hartree-Fock – approximation, they considered afictitious non-interacting system described by a set of single-particle Schrödinger-like equations:

[− ~2

2m0∇2 + VKS(r)

]ψk(r) = εnkψk(r) (1.14)

where the “effective Kohn-Sham potential ” VKS(r) must be chosen such that the value of the ground-state density ρ0(r) of the interacting system is accurately reproduced:

ρ0(r) =∑

εnk≤εF

|ψk(r)|2. (1.15)

The energy functional Eni[ρ(r)] of such a non-interacting system is explicitly known:

Eni[ρ(r)] = T0[ρ(r)] +

∫VKS(r)ρ(r)d

3r with T0[ρ(r)] =∑

εnk≤εF

〈ψk| −~2

2m0∇2

r|ψk〉. (1.16)

Moreover, it is formally possible to write the energy functional E[ρ(r)] of the initial interacting elec-tronic system (1.11) as follows:

E[ρ(r)] = T0[ρ(r)] +1

2

∫d3r

∫d3r′

e2

4πε0

ρ(r)ρ(r′)

|r− r′| + Exc[ρ(r)] +

∫V (r)ρ(r)d3r (1.17)

where Exc[ρ(r)] = G[ρ(r)]−T0[ρ(r)] is the “exchange-correlation energy” which contains all the many-body effects.

Consequently, in order that E[ρ(r)] and Eni[ρ(r)] can have the same ground-state density ρ0(r)– or the same global minimum under the same constraint of particle number Ne =

∫ρ(r)d3r –, the

condition on VKS(r) reads finally:

VKS(r) = V (r) + VH(r) + V xc(r) (1.18)

where VH(r) is the Hartree potential already defined in (1.3) – calculated with ρ0(r) – and V xc(r) isthe “exchange-correlation potential ” defined by

V xc(r) =δExc[ρ(r)]

δρ(r)

∣∣∣∣∣ρ0(r)

. (1.19)

kinetic energy T and electron-electron interaction Vee, taken over the class of wavefunctions that yield the density ρ(r):

F [ρ(r)] = minΨ→ρ(r)

< Ψ[ρ(r)]|T + Vee|Ψ[ρ(r)] > with T = − ~2

2m0

Ne∑

i=1

∇2ri

and Vee =e2

4πε0

Ne∑

i,j=1i<j

1

|ri − rj |. (1.12)


As already observed at the end of section 1.1, a “self-consistency condition” appears since theeffective potential depends on the density VKS(r) = VKS [ρ0(r)]. However, no approximation has beenintroduced so far by describing the interacting electronic system by such an effective model of non-interacting particles. Actually, the mapping between the initial many-body problem and the auxiliarynon-interacting effective system would be exact if an expression of the exchange-correlation potentialV xc(r) was known. Unfortunately, not only the expression of Exc[ρ(r)] but also a systematic seriesof approximation converging to the exact result is missing. An approximation on the form of theexchange-correlation potential V xc(r) must thus be made, in order to solve self-consistently the Kohn-Sham equations defined by (1.14) and (1.18).

1.2.3 Approximations to the exchange-correlation functional Exc[ρ(r)]

As we have previously said, the total energy functional (1.17) is known except for its exchange-correlation term Exc[ρ(r)]. Various approximations to the exchange-correlation functional have beendevised [32] but only the most popular of them will be described now.

The Local Density Approximation (LDA)

The most commonly adapted is the so-called “ local density approximation” (LDA), initially proposedin the original article of Kohn and Sham [93]:

ELDAxc [ρ(r)] =

∫ρ(r) εxc[ρ(r)] d

3r. (1.20)

In this expression, εxc[ρ] is the exchange-correlation energy per particle of the homogeneous electrongas evaluated with the density ρ. The approximation thus relies on the following postulate: theexchange-correlation energy associated to a particular density ρ(r) is “ locally” equal to the exchangecorrelation energy of an homogeneous electron gas, which has the same overall density as the initialdensity evaluated at the point r.

Although the exchange contribution εx[ρ] can be obtained analytically [48]:

εx[ρ] = −3

4

[3

πρ

] 13

(1.21)

the correlation contribution εc[ρ] remains unknown. In 1980, Ceperley and Adler performed a setof Quantum Monte Carlo (QMC) calculations for the homogeneous electron gas with different densi-ties [35]. They also provided a correlation energy parametrization based on this ground-state energyresults, which are nowadays of great use in the context of the local density mapping.

The Generalized Gradient Approximation (GGA)

The “generalized gradient approximation” (GGA) is as popular as the LDA, especially in quantumchemistry. Introduced in the late eighties, this approach refines the LDA by including the dependenceon the gradient of the density in the exchange-correlation energy per particle:

EGGAxc [ρ(r)] =

∫ρ(r) εxc[ρ(r),∇ρ(r)] d3r. (1.22)

Many choices for the parametrization of εxc are nowadays available in this framework, but we willnot describe them since their theoretical foundation is often not completely rigorous – or at best em-pirical. The interested reader can find more details about the exact properties of exchange-correlationenergy Exc[ρ(r)] and all the other existing approximations in the review [32].


The Local Spin Density Approximation (LSDA)

In many applications, going beyond the LDA is needed in order to include non-homogeneous spindensities, such as spin density wave instabilities, local spin moment formation and spin-orbit coupling.Therefore the density has to be resolved into its spin components ρ↑(r) and ρ↓(r). The DFT formula-tion based on these two fundamental variables2 has been developed in the mid seventies [62, 160], andthe LDA has then been extended to the “ local spin-density approximation” (LSDA).

The Hohenberg-Kohn theorems and the Kohn-Sham equations can be immediately rephrased, justby attaching a suitable spin-index to the densities. There are, however, some exceptions to this simplerule, and among them the construction of the exchange-correlation functional. For instance, it is knownthat the exchange energy must be calculated by:

ELSDAx [ρ↑(r), ρ↓(r)] =1

2

(ELDAx [2ρ↑(r)] + ELDAx [2ρ↓(r)]

)(1.24)

For the correlation energy however, no scaling relation of this type holds, so that ELSDAc [ρ↑(r), ρ↓(r)]is in practice either directly constructed in terms of the spin-densities or written by using, withoutformal justification, the same interpolation as here-above for the exchange energy [80, 160].

1.2.4 Success and limitations of DFT

Despite the approximation in the form for the exchange-correlation energy – which makes the theoryapplicable in practice –, DFT is widely used in the solid-state community. This is first of all becauseits implementations can yield astonishingly good results with respect to experiments for describingground-state properties of a wide class of materials. For instance, the lattice constants of simple crys-tals are obtained with an accuracy of about 1%, within the LDA [14]. Another advantage of DFT isdoubtlessly the high computational efficiency of its implementations.

Nevertheless, according to the Hohenberg-Kohn theorems, DFT is only a ground-state theory. Thisapproach provides no information on the excitation spectrum of the compounds, which is directly ob-served in experiments such as photo-emission spectroscopy, optics or transport. It is however commonnowadays to interpret the Kohn-Sham eigenvalues as the single-particle excitations of the system.This identification is strictly speaking unjustified since the Kohn-Sham eigenvalues are merely auxil-iary quantities, without any physical meaning. By using it, people actually do a tacit assumption: theelectrons in the physical system are indeed considered to be well-described by a model of independentparticles.

In practice, this procedure yields reasonable results in many solids but fails for some systems,like transition metals, transition metal oxides or rare-earth compounds, which contain open d andf -shells. Such materials are said to be “strongly correlated ”. In these cases, the electronic bandstructure obtained at the end of a DFT calculation and the true excitation spectrum of the compounddo not match, highlighting that their electrons can not be well-described with a single-particle picture.For some of them called “Mott insulators”, both the Kohn-Sham ground-state and spectrum are evenqualitatively wrong, since the insulating behavior of the material is not found and a metallic state ispredicted. The next sections are devoted to describe the main features of such compounds.

2The formalism can also be reformulated in terms of the total charge density ρ(r) and the spin-magnetization densitym(r):

ρ(r) = ρ↑(r) + ρ↓(r) and m(r) =e~

2m0c

[ρ↑(r)− ρ↓(r)

](1.23)


1.3 Electronic correlations

The methods presented so far have described the physical properties of electrons in solids by as-suming a single-particle picture. In the two previous sections, the electron-electron Coulomb in-teraction term of the general Hamiltonian (1.2) was indeed approximated by an effective potentialVeff(r) = VH(r) + V xc(r) felt independently by all electrons. However, Nature is – fortunately – farmore complicated: the “electronic correlations” can induce a huge range of physical phenomena, re-quiring new models and methods to describe them.

1.3.1 Introduction to strong correlations and Mott insulators

In many solids, the physical properties of electrons can be described – to a good approximation – byassuming an independent particle picture. This is particularly successful when one deals with broadenergy bands associated with a large value of the kinetic energy. In such cases, the valence electronsare “highly itinerant”: they are delocalized all over the solid. As a result, a description based on thenearly-free electron approximation is appropriate since the typical time spent by an electron near aspecific atom in the crystal lattice is very short.

For some materials however – generally associated with moderate values of the bandwidth –, adescription based on a particle-like picture – with a tight-binding model – appears then more suitablesince valence electrons spend a larger time around a given atom in the crystal lattice. In these condi-tions, electrons “have enough time to see each other on each atomic site” – via the Coulomb interaction– and the motions of individual electrons become “correlated ”. If the energy cost of this interactionis sufficiently large, delocalizing the valence electrons over the whole solid may become less favorableenergetically. In extreme cases, the electrons may even remain "localized" on their respective atom.

If this occurs to all electrons close to the Fermi level, the solid becomes an insulator. This insulatingstate is difficult to understand from a wave-like picture. It does not come from destructive interferencein k-space resulting in the absence of available one-electron states, as in conventional band insulators.However, this state is easily described with a particle-like picture in real space, as we have just done.This mechanism was explained long ago by Mott and Peierls [120, 121], that is why such compoundsare called “Mott insulators”.

The most interesting situation – but also the hardest to handle theoretically – arises when thelocalized character on short time-scales (or high energy scales) and the itinerant character on longtime-scales (or short energy scales) coexist. In such cases, the electrons can be naively seen as “hesi-tating” between being itinerant and being localized. Such materials are said to be “strongly correlated ”.

In these compounds, a plethora of physical phenomena can not be described by the standard bandtheory and DFT, as already mentioned in section 1.2. It is thus necessary to develop new techniques inorder to understand these properties. The most famous models in the domain were derived by Hubbardin 1963 [73] and 1964 [74, 75]. Despite their apparent simplicity, these “Hubbard models” embody wellthe necessity to think both in k-space and in real space to describe the physics of correlation.

1.3.2 The Hubbard model

For the sake of simplicity, we will derive here the “Hubbard model" for an electronic system in acrystal composed of a cubic lattice with one atom by unit cell. Moreover, it is more convenient to usethe formalism of “second quantization” to express the Hamiltonian (1.2). This consists in replacingthe wavefunction by an operator acting on a quantum field with a fluctuating particle number. As

1.3. ELECTRONIC CORRELATIONS 11

described in standard textbooks [17], the Hamiltonian (1.2) becomes in terms of field operators:

H =∑

σ=↑,↓

∫

R3

d3r ψ†σ(r)

[− ~2

2m0∇2

r + V (r)

]ψσ(r)

+1

2

∑

σ,τ=↑,↓

e2

4πε0

∫

R3

d3r

∫

R3

d3r′ ψ†σ(r)ψ

†τ (r

′)

[1

|r− r′|

]ψτ (r

′)ψσ(r)

(1.25)

where ψ†σ(r) and ψσ(r) respectively creates and annihilates an electron with spin σ at the point r. The

proper fermion statistics is imposed through the anticommutation relations of the field operators:

ψσ(r)ψτ (r′) + ψτ (r

′)ψσ(r) = 0 and ψσ(r)ψ†τ (r

′) + ψ†τ (r

′)ψσ(r) = δστδ(r− r′). (1.26)

It is generally of little use to start with the bare ion-electron potential V (r) and the long-rangeCoulomb interaction in (1.25), since a collective screening of the core and valence electrons may be large.Much of this screening effect can be incorporated into the single particle part of H by modifying locallythe potential3. As a result, the electron-electron interaction is renormalized in a residual interactionterm4:

Vee(r, r′) =

e2

4πε0

1

|r− r′| −1

Ne[Veff(r) + Veff(r

′)] (1.28)

with Veff(r) = VH(r) + V xc(r) and we can finally rewrite the Hamiltonian as:

H =∑

σ=↑,↓

∫

R3

d3r ψ†σ(r)

[− ~2

2m0∇2

r + VKS(r)

]ψσ(r)

+1

2

∑

σ,τ=↑,↓

∫

R3

d3r

∫

R3

d3r′ ψ†σ(r)ψ

†τ (r

′) Vee(r, r′) ψτ (r

′)ψσ(r).(1.29)

Introduced by Wannier in 1937 [163], the “Wannier states” provide a representation which is moreappropriate to our purpose. They are indeed localized around each atomic site Rj and defined as theFourier transformation of the Bloch states:

χRj

Lσ(r) =1√N∑

k

e−ik.RjψσkνL(r) (1.30)

where N is the number of lattice sites, L the combined index (l,m) denoting the orbital and νLthe band index associated to the L character5. Defining the creation and annihilation operator of aWannier state as cσ†RjL

and cσRjL, the field operators can be written as:

ψ†σ(r) =

∑

Rj ,L

[χRj

Lσ(r)]∗cσ†RjL

and ψσ(r) =∑

Rj ,L

χRj

Lσ(r)cσRjL. (1.31)

3This is precisely what is done with the effective Kohn-Sham potential VKS(r) in DFT where:

VKS(r) = V (r) + VH(r) + V xc(r) = V (r) + Veff(r). (1.27)

4In reality, screening is a dynamical process which involves collective charge fluctuations with a plasma frequencyscale. However, if the plasma frequency is higher than the excitation energies of interest – as in our case –, Vee(r, r

′) canbe taken as instantaneous.

5Contrary to the most general case presented in section 2.3, the definition for the Wannier states given here has noambiguity, since we are considering a very simple example.


The Hamiltonian becomes then the following lattice model:

H =∑

i,j

∑

L,L′

∑

σ

tLL′σ

ij cσ†RiLcσRjL′

+1

2

∑

i,j,k,l

∑

L,L′,M,M ′

∑

σ,σ′

ULL′MM ′σσ′

i j k l cσ†RiLcσ

′†RjL′c

σ′

RlM ′cσRkM

(1.32)

where the real-space hopping amplitudes tLL′σ

ij and the interaction parameters ULL′MM ′σσ′

i j k l are respec-tively given by:

tLL′σ

ij =

∫

R3

d3r[χRiLσ(r)

]∗[− ~2

2m0∇2

r + VKS(r)

]χRj

L′σ(r)

ULL′MM ′σσ′

i j k l =

∫

R3

∫

R3

d3rd3r′[χRiLσ(r)

]∗[χRj

L′σ′(r)]∗Vee(r, r

′) χRlM ′σ′(r)χ

RkMσ(r).

(1.33)

For the study of the basic physical mechanisms, simpler lattice model Hamiltonians are traditionallyused. They are derived from the general expression (1.32) by reducing the number of matrix elementsto the dominant contributions. For instance, to derive the single band [73] or the multi-orbital Hubbardmodels [74, 75], the hopping elements tLL

′σij are usually restricted to nearest-neighbor and next-nearest-

neighbor terms and the local intra-atomic interaction parameters ULL′MM ′σσ′

i i i i are expected to stronglydominate.

Consequently, the resulting Hamiltonian for the single-band Hubbard model reads:

H =∑

〈i,j〉,σ

tij cσ†i c

σj + U

∑

i

ni↑ni↓ + (ε0 − µ)∑

i,σ

cσ†i cσi (1.34)

where we used i instead of Ri and the operator niσ = cσ†i cσi to simplify the notations. Because of

the translational invariance of the system, the hopping amplitude tij is t for all nearest-neighbors iand j, t′ for all next-nearest-neighbors i and j, and 0 otherwise. Moreover, this one-particle part ofthe Hamiltonian (1.34) can be diagonalized in k-space. The chemical potential µ and the energy ε0 ofthe single-electron atomic level have been introduced to write the Hamiltonian in its more general form.

Using similar shortened notations, the multi-orbital Hubbard model reads:

H =∑

〈i,j〉,σ

∑

L,L′

tij cσ†iLc

σjL +

∑

i

∑

L,L′,σ,σ′

Uσσ′

LL′ niLσniL′σ′ +∑

i,L,σ

(εL − µ) cσ†iLcσiL (1.35)

where the on-site interaction term Uσσ′

LL′ is generally parametrized as follows:

∑

L

U nL↑nL↓ +∑

L>M

∑

σ

U ′ nLσnMσ + (U ′ − J) nLσnMσ

−∑

L 6=M

J[ψ†L↓ψ

†M↑ψM↓ψL↑ + ψ†

M↑ψ†M↓ψL↑ψL↓ + h.c.

]. (1.36)

Moreover, if one considers a lattice model restricted only to the t2g orbitals6 for each atom, the

6A brief reminder about the definition of the eg and t2g atomic orbitals can be found in Appendix A.

1.3. ELECTRONIC CORRELATIONS 13

parameter U ′ takes the value U − 2J and the multi-orbital Hubbard model can then be rewritten as:

H =∑

〈i,j〉,σ

∑

L,L′

tij cσ†iLc

σjL +

∑

i,L,σ

(εL − µ) cσ†iLcσiL

+∑

L

U nL↑nL↓ +∑

L>M

∑

σ

(U − 2J) nLσnMσ + (U − 3J) nLσnMσ (1.37)

−∑

L 6=M

J[ψ†L↓ψ

†M↑ψM↓ψL↑ + ψ†

M↑ψ†M↓ψL↑ψL↓ + h.c.

].

The Hamiltonian (1.37) with U and J has been constructed such that it is rotationally invariant bothin orbital and spin space. If we assume that the Hund’s coupling J is small compared to the on-siteCoulomb repulsion U , we can either set it to zero entirely or just set the exchange and pair-hoppingterms – of the third line – to zero. In both cases, the Hamiltonian will finally contain only density-density terms.

1.3.3 Examples of strongly correlated materials

Strongly correlated materials are generally associated with partially filled d or f -shells. As a result,these materials are made of:

• transition metal elements, particularly from the 3d-shell from Titane (Ti) to Copper (Cu), andto a lesser extent from the 4d-shell from Zirconium (Zi) to Silver (Ag),

• rare earth (4f -shell) from Lanthanum (La) to Ytterbium (Yb), or actinide elements (5f -shell)from Actinium (Ac) to Nobellium (No).

In the following, we briefly describe the key issues arising in these compounds and give a few repre-sentative examples in each case. The interested reader can find in reviews [55, 66, 76] more importantdiscussions on this topic.

Transition metals

In 3d-transition metals, the 4s orbitals have lower energy than the 3d and are thus filled first. Moreover,the 4s orbitals extend much further from the nucleus and overlap strongly. This holds the atomssufficiently so that the 3d orbitals have a small direct overlap, hence forming band neither extremelynarrow nor really wide.

In addition, the 3d orbital wavefunctions are confined closer to the nucleus than the other orbitals.They then undergo an efficient Coulomb repulsion, despite the screening of the 4s orbitals. For thesetwo reasons, electron correlations do have important physical effects for 3d-transition metals, but notextreme ones leading to a complete localization.

Consequently, band structure calculations based on DFT-LDA methods generally overestimate thewidth of the occupied d-band – by about 30% in the case of Nickel (Ni), for instance [55]. Furthermore,some features observed in spectroscopy experiments – such as the 6 eV satellite in Ni – are alsosignatures of correlation effects, and are not reproduced by standard electronic structure calculations.

Transition metal oxides

In transition metal compounds – oxides or chalcogenides –, the direct overlap between d-orbitals isgenerally so small that d electrons can only move through hybridization with the ligand atoms. Thisleads then to the formation of quite narrow bands and the correlation can then play an even moresignificant role than in simple transition metals.

As a result, these systems can even turn into Mott insulator, like the d1-compounds lanthanumtitanate (LaTiO3) or yttrium titanate (YTiO3) [130]. For such system, band structure calculations


based on DFT-LDA methods completely fail as a metallic behavior is always predicted. For the samereason, DFT calculations also failed in describing the metal-insulator transition in compounds likevanadium dioxide (VO2) and vanadium sesquioxide (V2O3) [158].

But even for metallic transition metal oxides, the LDA band structure can be found to disagreewith experimental observations: this is for instance the case for strongly correlated metal strontiumvanadate (SrVO3) [101].

Rare earth and actinides

A distinctive character of the physics of rare-earth metals is that the 4f electrons tend to be localizedrather than itinerant at ambient pressure. The f -electrons then contribute little to the cohesive energyof the solid, and the unit cell volume depends very weakly on the filling of the 4f -shell.

For these materials, ground-state properties, such as equilibrium unit cell volume, are not accu-rately predicted from LDA calculations. A spectacular example is the δ-phase of metallic Plutonium(Pu) in which the unit cell volume is underestimated (compared to the experimental value) by as muchas 35% [55].

All these examples illustrate the need of a a method which is able to handle intermediate situationsbetween “fully localized” and “fully itinerant” electrons. The “dynamical mean-field theory” (DMFT)is a technique which was developed in the early nineties with this aim in view.

1.4 The Dynamical Mean-Field Theory (DMFT)

In this section, we introduce the “dynamical mean-field theory” (DMFT) [57, 98], which can be seenas an extension to quantum many-body systems of classical mean-field approach. This theory hasbeen a major breakthrough in the understanding of correlated materials since it allowed a consistentdescription of both the low-energy coherent features – the long-lived quasiparticle excitations – andthe incoherent high-energy excitations due to the Coulomb repulsion, acting on short timescales.

In the following, we first present the main features of this quantitative method, assuming that thereader is familiar with the fundamentals of many-body physics and Landau’s theory of Fermi liquid.We then provide a general description of the Mott transition within the DMFT framework by basingour discussion on the example of the half-filled one-band Hubbard model (1.34) on a Bethe lattice ininfinite dimensions.

1.4.1 Introduction to the theory

The underlying physical idea in a mean-field approach of a lattice problem is that the dynamics at agiven site of the can be understood as the interaction of the local degrees of freedom at this site withan “external bath” created by all other degrees of freedom on the other sites. To make it simple, oneobtains the “dynamical mean-field theory” (DMFT) by merely applying this idea to the quantum case.

Historically, the most important steps leading to this quantum generalization were the introductionof the limit of large lattice coordination for interacting fermion models in 1989 by Müller-Hartmann[123] and Metzner and Vollhardt [113] and the mapping of the reference system onto a self-consistentquantum impurity model in 1992 by Georges and Kotliar [56].

We derive here DMFT equations on the simplest example of the single-band Hubbard model:

H =∑

〈i,j〉,σ

tij cσ†i c

σj + U

∑

i

ni↑ni↓ + (ε0 − µ)∑

i,σ

cσ†i cσi . (1.34)

1.4. THE DYNAMICAL MEAN-FIELD THEORY (DMFT) 15

With this Hamiltonian and in the absence of hopping tij = 0, each atom has four eigenstates:

• |0〉 with energy 0,

• | ↑〉 and | ↓〉, both with energy ε0 − µ,

• | ↑↓〉 with energy 2(ε0 − µ) + U .

The key quantity on which DMFT focuses is the local Green’s function at a given lattice site i,defined by:

∀τ, τ ′ ∀σ =↑, ↓ Gσii(τ − τ ′) = −〈T [cσi (τ)cσ†i (τ ′)]〉. (1.38)

Because of the translational invariance of the system, this quantity is the same for all site i and we willmerely refer to it as Gσloc(τ − τ ′) in the following. In a completely similar manner as in the classicalmean-field theory, one then introduces a representation of the local Green’s function as a function of“a single atom coupled to an effective bath”. This step can be performed by using the Hamiltonian ofan Anderson impurity model (AIM) [7]:

HAIM = Hatom +Hbath +Hcoupling (1.39)

where Hatom describes the local energy on the “atomic (or impurity) site”:

Hatom = U ni↑ni↓ + (ε0 − µ)(ni↑ + ni↓) (1.40)

a set of non-interacting fermions (described by the field operators a†lσ and alσ) has been introduced todescribe the “effective external bath”:

Hbath =∑

lσ

εl a†lσalσ (1.41)

and Hcoupling describes the processes of bath fermions hopping on or off the atomic site with ampli-tude Vl:

Hcoupling =∑

lσ

Vl (a†lσc

σi + cσ†i alσ) (1.42)

The parameters εl and Vl should be chosen in such a way that the impurity Green’s function Gσimp(τ−τ ′)of (1.39) coincides with the local Green’s function Gσloc(τ − τ ′) of the lattice Hubbard model underconsideration:

∀τ, τ ′ ∀σ =↑, ↓ Gσimp(τ − τ ′) = Gσloc(τ − τ ′). (1.43)

They also enable to define an hybridization function:

∆(iωn) =∑

l

|Vl|2iωn − εl

. (1.44)

This is easily seen when the effective on-site problem is recast in a form which does not explicitlyinvolves the effective bath degrees of freedom. However, this requires the use of an effective action Seff

based on the functional integral formalism rather than a simple Hamiltonian formalism. Integratingout the bath degrees of freedom, this effective action can indeed be written as:

Seff = −∫ β

0dτ

∫ β

0dτ ′∑

σ

cσ†i (τ)G−10 (τ − τ ′)cσi (τ

′) + U

∫ β

0dτ ni↑ni↓ (1.45)

where cσ†i and cσi are the Grassmann variables corresponding to the local “atomic” state and

G−10 (iωn) = iωn + µ− ε0 −∆(iωn). (1.46)


In expression (1.45), G0 plays the role of a bare Green’s function for the effective action Seff, but itshould not be confused with the non-interacting local Green’s function of the original lattice model(obtained for U = 0). Moreover, one can interpret G−1

0 (τ − τ ′) – or equivalently ∆(iωn) – as thequantum generalization of the Weiss effective field in the classical case. The main difference with theclassical case is that this “dynamical mean-field ” is a function of energy – or time – instead of a singlenumber.

The local action (1.45) represents the effective dynamics of the local site under consideration: afermion is created on the site i at time τ (coming from the external bath, in other words from theother sites of the lattice) and is destroyed at time τ ′ (going back to the bath). Whenever two fermions– with opposite spins – are present at the same time, an energy cost U is included. This effectiveaction thus describes the fluctuations between the four atomic states |0〉, | ↑〉, | ↓〉 and | ↑↓〉 inducedby the coupling to the bath, as displayed on figure 1.1. Taking full account of these local quantumfluctuations is precisely the main purpose of DMFT.

Figure 1.1: DMFT enables to describe the fluctuations on a lattice site between the four atomic states|0〉, | ↑〉, | ↓〉 and | ↑↓〉 induced by the coupling to an external effective bath. From [99].

So far, we have introduced the quantum generalization of the Weiss effective field and have repre-sented the local Green’s function Gloc(τ − τ ′) as that of a single atom coupled to an effective bath.This can be viewed as an exact representation. We now have to generalize to the quantum case themean-field approximation relating the Weiss function G−1

0 (τ−τ ′) to Gloc(τ−τ ′). The simplest mannerto explain it is to observe that, in the effective impurity model (1.45), we can define a local self-energyfrom the interacting Green’s function Gσimp(τ − τ ′) = −〈T [cσi (τ)cσ†i (τ ′)]〉Seff

and the Weiss dynamicalmean-field as:

Σimp(iωn) = G−10 (iωn)−G−1

imp(iωn) = iωn + µ− ε0 −∆(iωn)−G−1imp(iωn). (1.47)

On the contrary, the self-energy of the original lattice model can be defined as usual from the fullGreen’s function Gσi−j(τ − τ ′) = −〈T [cσi (τ)cσ†j (τ ′)]〉 by:

G(k, iωn) =1

iωn + µ− ε0 − εk −Σ(k, iωn)(1.48)

in which εk is the Fourier transform of the hopping integral:

εk =∑

j

tijeik·(Ri−Rj). (1.49)


The “DMFT approximation” then consists in identifying the lattice self-energy and the impurityself-energy. In real-space, this means that all non-local components Σij(iωn) are neglected and theon-site components are approximated by Σimp(iωn):

∀iωn ∀i Σii(iωn) ≈ Σimp(iωn) and ∀i, j Σij(iωn) ≈ 0. (1.50)

Summing (1.48) over k in order to obtain the on-site component Gloc(iωn) of the the lattice Green’sfunction, and using (1.47), one finally gets the “self-consistency condition”:

∑

k

1

iωn + µ− ε0 − εk +G−1(iωn)− G−10 (iωn)

= G(iωn) (1.51)

or by considering explicitly the hybridization function:

∑

k

1

∆(iωn) +G−1(iωn)− εk= G(iωn). (1.52)

The expression (1.52) can also be written as:

∫dε

D(ε)

∆(iωn) +G−1(iωn)− ε= G(iωn) with D(ε) =

∑

k

δ(ε− εk). (1.53)

where D(ε) is the non-interacting density of states.

This self-consistency condition relates, for each frequency, the dynamical mean-field ∆(iωn) – orG0(iωn) – and the local Green’s function G(iωn). Furthermore, G(iωn) is also the interacting Green’sfunction of the effective impurity model (1.39) – or (1.45) – since by construction:

∀iωn Gloc(iωn) = Gimp(iωn) = G(iωn). (1.43)

Therefore, the set of DMFT equations is closed and fully determines in principle the two functions ∆and G – or G0 and G.

In practice, one uses an iterative procedure based on a self-consistency loop to solve this problem.In many cases, this iterative procedure converges to a unique solution independently of the initialchoice of ∆(iωn). In some cases however, more than one stable solution can be found, especially closeto the Mott transition.

1.4.2 Limits in which DMFT becomes exact

The DMFT equations yield the exact answer in three simple limits:

Infinite coordination limit

As the classical mean-field theory, DMFT becomes exact in the limit where the connectivity z of thelattice is taken to infinity. Apart from the intrinsic interest of solving strongly correlated fermionmodels in the limit of infinite coordination, this property guarantees that exact constraints – such asthe causality of the self-energy, the non-negativity of the spectral functions, the Luttinger theorem orother sum rules – are preserved by the DMFT approximation.


Non-interacting limit

In the non-interacting limit – that is to say when U = 0 in (1.34) –, the DMFT approximation (1.50) istrivially exact since the self-energy is not only k-independent but merely vanishes. As a result, solvingthe impurity problem (1.45) yields G(iωn) = G0(iωn) and, by using the condition (1.51), G(iωn)reduces to the free on-site Green’s function, as expected:

∀ σ =↑, ↓ Gσ(iωn) =∑

k

1

iωn + µ− ε0 − εk. (1.54)

Atomic limit

In the atomic limit – that is to say when tij = 0 in (1.34) –, the model consists in a collection ofisolated atoms on each site. As a result, the DMFT approximation is again exact since the self-energyhas only on-site components. In this case, the dynamical mean field G0(iωn) vanishes in (1.45) andthe action Seff merely corresponds to the quantization of the atomic Hamiltonian Hatom (1.40). Thisyields:

∀ σ =↑, ↓ Gσ(iωn) =(1− n

2

) 1

iωn + µ− ε0+n

2

1

iωn + µ− ε0 − U

and Σσ(iωn) =n

2U +

n

2

(1− n

2

) U2

iωn + µ− ε0 − (1− n2 )U

withn

2=

eβ(µ−ε0) + eβ(2(µ−ε0)+U)

1 + 2eβ(µ−ε0) + eβ(2(µ−ε0)+U).

(1.55)

Thus, besides the case of infinite dimensions, DMFT is exact in the two limits of the non-interactingband and of the isolated atoms and can then be seen as providing an interpolation between them.

1.4.3 Impurity solvers

Using reliable methods for calculating the impurity Green’s function and self-energy is a key step insolving the DMFT equations. “Impurity solvers” can be classified into two main types:

• the analytical methods, like the iterated perturbation theory (IPT) [56, 81] or the non-crossingapproximation (NCA) [98],

• the numerical techniques, such as the exact diagonalization [30, 138], the renormalization group(NRG) [28] and the quantum Monte Carlo algorithms (QMC) [69, 141, 165].

We provide here only a short introduction to the QMC methods since our calculations on strontiumiridate (Sr2IrO4) were performed by using a continuous time QMC (CTQMC) technique [141, 165]. Amore thorough description and analysis of impurity solvers can be found in [57].

The major advantage of QMC solvers over other impurity solvers is that they are numerically exact,easily adapted to multiple orbital systems or clusters, and fast enough to reach low temperatures.However, their main disadvantage is that they work on the imaginary time (or Matsubara axis).Real frequency data like spectra or optical conductivities have thus to be extracted via analyticalcontinuation [78]. Two main QMC algorithms are in wide use in the community:

The algorithm of Hirsch and Fye [69] was developed long before DMFT as an algorithm to solvethe Anderson impurity model. It was the first QMC algorithm applied to the DMFT impurity


problem and is still very popular. This algorithm is based on a Trotter-Suzuki decomposition ofthe effective action Seff (1.45) and a discrete Hubbard-Stratonovich transformation [67, 68]. Ittherefore requires a discretization of imaginary time into N so-called “time slices” ∆τ = β/N .

The CTQMC algorithms: In 2005 Rubtsov et al. [141] introduced the “weak-coupling CTQMCalgorithm” for fermions and in 2006 Werner et al. [165] the “strong-coupling CTQMC algorithm”.The interested reader can find a detailed presentation of these techniques in [61, 63]. Both thesemethods have the main advantage of relying on continuous time. As a result, they are free ofany systematic error due to a time discretization and can resolve accurately the behavior of theGreen’s function for lower temperatures and stronger interaction regime than the Hirsch-Fyealgorithm7. For these main reasons, CTQMC methods are steadily gaining importance in thefield.

1.4.4 Theory of the Mott transition within DMFT

By applying DMFT on the Hubbard model, huge progress in the understanding of the Mott transitionhave been made. In the following, we briefly review the key features of this phenomenon. More detailson the Mott metal-insulator transition are available in the review [57] and in the following originalreferences [56, 58, 59, 100, 139, 140].

To illustrate our discussion, we will consider the half-filled one-band Hubbard model (1.34) on aBethe lattice in infinite dimensions. In this case, the density of states (DOS) of the band is semi-circular, as depicted in figure 1.2. The schematized phase diagram of this model represented as afunction of the interaction U and the temperature T is shown in figure 1.3.

-1.5 -1 -0.5 0 0.5 1 1.5Energy (eV)

0

0.1

0.2

0.3

0.4

0.5

0.6

DO

S (s

tate

s.eV

-1)

Figure 1.2: Density of states of the Bethe lattice in infinite dimensions, with an half bandwidthD = 1.0 eV.

Non-interacting limit

As already mentioned, in the non-interacting limit (U = 0), the Hamiltonian (1.34) is diagonal ink-space. As a result, the spectral function of the system at a given momentum is a Dirac distribution:

A(k, ω) = − 1

πIm [G(k, ω)] = δ(ω − εk) (1.56)

7Indeed, the computation time of Hirsch-Fye algorithm approximately scales as O(N3) whereas its grid spacing isgenerally required to be N ≈ 5 βU to ensure a sufficient resolution.


Figure 1.3: Schematic phase diagram of the one band half-filled Hubbard model (1.34) within DMFT.The axes correspond to the temperature T and the Coulomb repulsion U . Energies are normalized bythe bandwidth of the model W = 2D. From [99].

and the self-energy is of course zero. The total local spectral function A(ω) =∑

kA(k, ω) coincideswith the density of states shown in figure 1.2.

The Fermi liquid regime

At low temperature and moderate interaction strength, the system exhibits the “Fermi liquid regime”.The DMFT spectral function A(ω) – at T=0 K – presented in figure 1.4-a displays the famous “three-peak structure”, made of a quasi-particle band close to the Fermi energy surrounded by the lowerand upper Hubbard bands. The quasi-particle part has a reduced bandwidth of order Z.D ∼ ε∗F ,where D is the half-bandwidth of the non-interacting initial system and Z the quasi-particle weight.This energy scale can also be interpreted as the coherence-scale for the quasi-particles. Moreover,the value A(ω = 0) is pinned to its non-interacting value, which means that the correlations do notmodify the Fermi surface [124]. In addition to low energy quasi-particles, the preformed Hubbardbands accommodate the weight 1−Z which is transferred from lower energies. They can be associatedto the atomic-like transitions corresponding to the addition or removal of one electron on an atomic site.

When looking at the corresponding self-energy presented in figure 1.4-b, the characteristics ofthe Fermi liquid are present: the real-part is linear in frequency, with a negative slope, around theFermi-level and the imaginary part is proportional to ω2. However, already at rather low energies,the self-energy deviates substantially from its low-energy behavior. Furthermore, the real-part has thesame behavior at high energy as the atomic limit: Σ(ω → +∞) ≈ U2/4ω and approaches the constantHartree term, which is zero at half-filling. The matching of these two very different behaviors resultsin a pronounced frequency dependence in the intermediate regime, leading to the observed prominentpeak in the imaginary part.

An increase in temperature induces a finite scattering rate even at the Fermi level. This scatteringrate corresponds to a finite value in the imaginary part of the self-energy at zero energy and resultsinto the violation of the previously mentioned pinning condition: the spectral function at zero energyis no longer bound to its non-interacting value, but is considerably reduced.


-4 -3 -2 -1 0 1 2 3 4Energy (eV)

0

0.1

0.2

0.3

0.4

0.5

0.6

DO

S (s

tate

s.eV

-1)

initial DOSU = 2 eV

(a) - Spectral function in the Fermi liquid regime

-4 -3 -2 -1 0 1 2 3 4Energy ω (in eV)

-2

-1

0

1

2

Re[ Σ(ω) ]Im[ Σ(ω) ]

(b) - Self energy in the Fermi liquid regime

Figure 1.4: DMFT spectral function A(ω) (panel a) and self-energy Σ(ω) (panel b) for the one bandhalf-filled Hubbard model (1.34) on the Bethe lattice in infinite dimensions with U=2 eV. The calcu-lations were performed at T=0 K with the iterated perturbation theory (IPT).

-4 -3 -2 -1 0 1 2 3 4Energy (eV)

0

0.1

0.2

0.3

0.4

0.5

0.6

DO

S (s

tate

s.eV

-1)

initial DOSU = 4 eV

(a) - Spectral function in the Mott insulating regime

-4 -3 -2 -1 0 1 2 3 4Energy ω (in eV)

-8

-6

-4

-2

0

2

4

6

8

Re[ Σ(ω) ]Im[ Σ(ω) ]

(b) - Self energy in the Mott insulating regime

Figure 1.5: Same as figure 1.4, but with U=4 eV.


The Mott insulating state

At strong enough coupling, the system becomes a “Mott insulator ”. The gap ∆ ≈ U is indeed clearlyseen on the corresponding DMFT spectral function A(ω) – at T=0 K – displayed in figure 1.5-a. Sinceat half-filling, the real part of the self-energy is an anti-symmetric function, it has to vanish at zerofrequency. As a consequence the elimination of spectral weight at the Fermi level can only be achievedby a divergence of the imaginary part, as observed on figure 1.5-b.

In addition, this insulating phase is characterized by unscreened local moments, associated with aCurie law for the local susceptibility

∑q χq ≈ 1

T and an extensive entropy. As temperature is lowered,these local moments order into an antiferromagnetic phase. On the contrary, upon heating, thermalfluctuations lead to a filling of the gap and a curing of the divergence in the imaginary part of theself-energy. The latter thus reaches a finite value at zero frequency.

The transition lines

The Mott metal-insulator transition is of first order, and thus exhibits a coexistence region in the phasediagram, as seen on figure 1.3. At T = 0 K, the mean-field solution corresponding to the paramagneticmetal indeed disappears at a critical coupling Uc2 whereas the mean-field insulating solution is foundfor U > Uc1. More precisely, at the point Uc2 the quasi-particle weight vanishes (Z ≈ 1 − U/Uc2) asin the Brinkman-Rice theory [27] while the Mott gap ∆ opens up at Uc1. For the half-filled one-bandHubbard model, the value for Uc1 and Uc2 at T=0 K are 3 D and 2.5 D respectively.

These two critical couplings extend at finite temperature into two spinodal lines Uc1(T ) and Uc2(T )and end in a critical point, above which the transition from a high-conductivity to the high-resistivityphase is continuous. The phase diagram is thus similar to the one of the liquid-gas transition, castingthe metal-insulator Mott transition into the Ising universality class [96, 97].

In a realistic multi-band setup, the situation becomes more involved [133]. In particular, a modi-fication of the Fermi surface becomes possible through a local self-energy that causes charge transfersbetween different orbitals. Moreover the necessity of a divergent mass for a Mott transition gets re-laxed. Indeed a correlation enhancement of crystal field splittings may cause a shifting of spectralweight and thus lead to the separation of former bands at the Fermi level. Responsible for this will bethe orbital dependence in the real-parts in the self-energy, which need no longer vanish at zero energy,as was the case for the half-filled one-band model.

Chapter 2

Combining DFT-LDA calculations withDMFT: the LDA+DMFT approach

As emphasized in sections 1.2 and 1.3, the screened Coulomb interaction is responsible for the failureof DFT-LDA calculations in giving a reliable independent-particle description of correlated materials.Indeed, in these systems, the band formation – well-described in the k-space – and the tendency tolocalization – rather described in real space – coexist and a theoretical framework in which these bothissues are treated on an equal footing is necessary.

The dynamical mean-field theory (DMFT), introduced in section 1.4, is a quantitative methodwhich was precisely developed in the early nineties for handling electron correlations. Since then, ithas led to significant advances in the understanding of strong correlation physics, especially by allowingto describe the Mott metal-insulator transition.

In order to overcome the shortcomings of DFT-LDA for strongly correlated materials, the com-bination of the electronic band structure techniques with DMFT was suggested [57] and the firstdescriptions of the so-called “LDA+DMFT formalism” appeared in 1997-1998 [11, 104]. The successof this approach was such that numerous implementations have been developed and applied to thecalculations of the spectral properties of many materials over the past decades [98].

In this chapter, we first introduce the key steps involved in performing an LDA+DMFT calculation.This description is meant to be as general as possible, irrespective of any specific basis set, any bandstructure code or any impurity solver chosen to implement the method. On the contrary, the followingsections focus on the technicalities of the implementation we have extended to perform our calculationson Sr2IrO4: this implementation is based on the “ linear augmented plane waves (LAPW) approach” –and more precisely on the Wien2k program – and uses a “projection scheme to localized Wannier-typeorbitals” [1] .

2.1 The LDA+DMFT formalism

2.1.1 General description of the method

The “LDA+DMFT approach” [104, 11] to electronic structure is based on the following “philosophy” :one extracts from an initial LDA calculations the part of the system which is assumed to exhibit themost significant correlation effects in order to treat it within DMFT, whereas the other part – consid-ered as less correlated – is assumed well-described by the standard LDA approach. Consequently, itsstarting point is very similar to that of the “LDA+U approach” [13, 9], which combines LDA calcula-tions with a static repulsion U .

23

24 CHAPTER 2. COMBINING DFT-LDA CALCULATIONS WITH DMFT

However, DFT and DMFT rely on two different key quantities. On the one hand, a DFT cycleis built around the total electronic charge density ρ(r) of the system but gives a description of theelectronic wavefunctions based on the Bloch basis |ψσkν〉, with ν the band index, σ the spin degree offreedom and k the momentum (cf. section 1.2). On the other hand, DMFT can be described as aneffective atom approach and is based on the self-consistently determination of the local one-particleGreen’s function Gloc(iωn) of the system.

In the LDA+DMFT formalism, both these quantities – namely ρ(r) and Gloc(iωn) – must bedetermined self-consistently, following an iterative cycle which is summarized on figure 2.1. The presentdescription of LDA+DMFT follows this cycle, starting from a given charge density ρ(r):

i) The DFT calculation

The first step is merely based on the Kohn-Sham equations introduced in section 1.2. A Kohn-Shampotential VKS [ρ(r)] is indeed found and the related one-particle Hamiltonian is diagonalized:

H|ψσkν〉 = εσkν |ψσkν〉. (2.1)

Since the Bloch basis |ψσkν〉 is a natural output of any electronic structure calculation, we will choose

it in the following as the complete basis set to describe the full Hilbert space of the system.

ii) The “projection operators”

In order to formulate the local effective atomic problem for DMFT, one must build a set of local-ized basis functions |χαL〉, where α labels an atom in the unit cell and L stands for all orbital indicesL = l,m, σ. These functions will span the “correlated ” subspace C of the full Hilbert space, in which

many-body correlation effects – beyond the LDA – will be taken into account.

This set of orbitals |χαL〉 is seldom the same as the complete basis set used to expand the Kohn-Sham eigenstates |ψσkν〉 in the band structure code. The way to construct them is highly dependenton the implementation used and we will come back to this point later. For the current description, itis enough to introduce the following “projection operator ” onto the subspace C:

P(C)α =

∑

|χαL〉∈C

|χαL〉〈χαL|. (2.2)

Building this projector corresponds to the second step of the cycle, called “interfacing” on figure 2.1.For simplicity, we will consider in the following that only a single type of atom is included in thisprojection, or in other words is correlated.

iii) The DMFT self-consistent loop

Once defined the correlated subspace C, it is possible to construct the effective impurity model, definedby the bare Green’s function – or dynamical mean-field – [G0]

σLL′(iωn) and the many-body interaction

term HU (1.36). Its action Seff reads:

Seff = −∫ β

0dτ

∫ β

0dτ ′∑

LL′

c†L(τ)[G−10

]LL′ (τ − τ ′)cL′(τ ′) +

∫ β

0dτHU (c

†L; cL) (2.3)

where c†L and cL are the Grassmann variables corresponding to the correlated orbitals |χL〉.

2.1. THE LDA+DMFT FORMALISM 25

Figure 2.1: The complete self-consistent loop for LDA+DMFT:

The charge ρ(r) determines the Kohn-Sham potential from which the eigenvalues εσkν and Blochstates |ψσkν〉 are calculated (i).

The correlated orbitals are then defined and their projector P(C)α are constructed (ii) in order to

perform the DMFT loop. The latter (iii) consists in:- solving the effective impurity problem for the impurity Green’s function Gimp, hence obtaining animpurity self-energy Σimp;- combining the self-energy correction with the Green’s function of the solid G(iωn) in order tocalculate the local Green’s function Gα

loc(iωn) – cf. equations (2.6),(2.7) and (2.8) –;- finally obtaining an updated dynamical mean-field G0 for the impurity problem.Once the DMFT loop has converged, the chemical potential is updated and the new charge density –including many-body effects – is constructed (iv).


As explained in section 1.4, the dynamical mean-field G0 is the relevant link in the inner self-consistent DMFT loop. Starting from the following initial guess:

G0(iωn) = P(C)α GKS(iωn)P

(C)α with [GKS(k, iωn)]

σνν′ =

1

iωn + µ− εσkνδνν′ (2.4)

the effective impurity model is solved by using a suitable impurity solver. One then obtains theimpurity Green’s function Gimp(iωn) as well as the impurity self-energy:

[Σimp(iωn)

]LL′ =

[G−10 (iωn)

]LL′ −

[G−1imp(iωn)

]LL′ ∀ |χαL〉 , |χαL′〉 ∈ C. (2.5)

In order to have a full set of self-consistent equations, the effective impurity problem is related tothe whole solid thanks to the DMFT approximation (1.50). It states that the self-energy correctionin the solid must be non-zero only inside the (lattice-translated) correlated subspace and moreoverexhibits only local components in the basis set |χαL〉. The lattice self-energy correction Σσνν′(k, iωn) istherefore obtained by the following relation:

Σσνν′(k, iωn) = 〈ψσk,ν |P(C)α

[∆Σα

imp(iωn)]LL′P

(C)α |ψσ

k,ν′〉(2.6)

with[∆Σα

imp(iωn)]LL′ =

[Σimp(iωn)

]LL′ −

[Σdc

]LL′ .

In this expression, Σdc is the “double-counting term” for the local orbitals, which correct for correlationeffects already included in conventional DFT.

Using then Dyson’s equation, the Green’s function of the solid G(k, iωn) is given by

[Gσ(k, iωn)]−1νν′ = (iωn + µ− εσkν)δνν′ − Σσνν′(k, iωn) (2.7)

and the local Green’s function Gαloc is then obtained by projecting the Green’s function of the solid to

the set of correlated orbitals L of the correlated atom α:

Gαloc(iωn) = P(C)

α G(iωn)P(C)α . (2.8)

Finally, since by construction the local Green’s function must coincide with the one obtained from theeffective impurity problem:

Gαloc(iωn) = Gimp(iωn) (2.9)

a new dynamical mean-field G0 can be found thanks to the self-energy of the impurity model:[G−10 (iωn)

]LL′ =

[Σimp(iωn)

]LL′ +

[Gαloc(iωn)

−1]LL′ . (2.10)

This new G−10 allow to solve a new effective impurity model and the cycle is repeated until convergence

is reached.

iv) The updating of the charge

The last step of the LDA+DMFT loop is the “updating of the charge” ρ(r). Without introducing anyspecific basis set, the charge density is calculated from the full Green’s function of the solid by:

ρ(r) =1

β

∑

n

〈r|G(iωn)|r〉eiωn.0+ . (2.11)

In this expression, it is more convenient to split ρ(r) as follows:

ρ(r) = ρKS(r) + ∆ρ(r) with ρKS(r) = 〈r|GKS |r〉 =∑

k,ν,σεσk,ν≤µ

|ψσk,ν |2. (2.12)


It is important to point out here that the demand for charge neutrality is not imposed on ρKS(r) butrather on ρ(r). As a result, the chemical potential µ must be explicitly determined at the end of aDMFT loop in such a way that the total number of electrons Ne is the correct one:

Ne = tr(G) =

∫drρ(r). (2.13)

This value of the chemical potential will in general not be such that tr(GKS) =∫drρKS(r) = Ne. This

is actually not surprising: since correlation effects were introduced, the Kohn Sham representation ofthe charge density by independent Kohn Sham wave-functions no longer holds.

With this new charge density ρ(r) a new Kohn-Sham potential can be determined, and the wholecycle can be iterated again, until the charge density ρ(r), the impurity self-energy Σimp(iωn) and thechemical potential µ are converged.

However, in practice, the calculations using the self-consistency over the charge density in theLDA+DMFT framework still remains rare. So far, this has been implemented only within the linearmuffin-tin orbital (LMTO) framework [134, 142, 143] or within the Korringa-Kohn-Rostoker (KKR)method [114]. Very recently, an implementation based on the projector-augmented wave (PAW) for-malism was also presented [60]. In most cases however, the global self-consistency on the charge densityis not implemented. The LDA+DMFT calculations are thus performed by starting from the alreadyconverged density ρ(r) obtained at the DFT-LDA level and by then iterating the DMFT loop untilconvergence of the impurity self-energy. The LDA+DMFT calculations performed on Sr2IrO4 in thisthesis were done within this “one-shot” approach.

2.1.2 The double-counting correction

Since electronic correlations are already – but partially – taken into account within the DFT approachthrough the LDA – or GGA – exchange-correlation potential, the double-counting correction Σdc mustcorrect for this in the LDA+DMFT method. However, defining the double counting correction isactually a tricky problem in the framework of conventional DFT [42, 132] since this theory is notorbitally resolved. Furthermore the LDA – or even GGA – exchange-correlation potential does nothave a diagrammatic interpretation which would allow to subtract the corresponding terms from theDMFT many-body correction.

One could then think of subtracting the matrix elements of VH(r) and V xc(r) in the orbitals ofthe correlated subspace C from the Green’s function GKS(iωn) to which the many-body self-energyis applied. However, this option is not chosen because the DMFT approach is meant to treat thelow-energy screened interaction. As a result, the Hartree approximation is not an appropriate startingpoint and one wants to benefit from the spatially resolved screening effects which are already partiallycaptured in the LDA description of the system.

Systematic approaches to avoid the double-counting problem are still being developed. Neverthe-less, various schemes for the double counting correction currently exist. Among them, the followingare the most used:

• the “fully localized limit”, originally introduced in the LDA+U context [12]:

[Σdc

]σmm′ =

[U

(Nc −

1

2

)− J

(Nσc − 1

2

)]δmm′ . (2.14)

In this expression, U is the average Coulomb interaction, J the Hund’s coupling, Nσc the spin-

resolved occupancy of the correlated orbital and Nc = N↑c +N

↓c . As usual, the parameters m ans

σ refer to the orbital index and the spin degree of freedom respectively.


• the “around mean-field ” correction, also originally introduced in the LDA+U context [13]:

[Σdc

]σmm′ =

U

2Nc(Nc − 1)δmm′ . (2.15)

• the recent “Held’s convention”, which is specially adapted for a description where only the t2gorbitals are considered correlated on an atom [66]:

[Σdc

]σmm′ = (U − 2J)(Nc −

1

2)δmm′ . (2.16)

• the “Lichtenstein’s correction”, which was suggested for metallic systems since the static part ofthe correlation effects are already well described in DFT[105]:

[Σdc

]σmm′ =

1

2Tr[Σ(ω = 0)]. (2.17)

A recent work [83] has systematically investigated the effects of these different choices on thespectral function.

2.1.3 Choice of the localized basis set

In LDA+DMFT, the main physical issue is to construct the localized basis set |χαL〉 and define amongthem the correlated orbitals of the system. As observed in section 1.3, a suitable concept is that of“Wannier functions” (1.30), formally built from the Bloch basis |ψσ

k,ν〉 and centered on the atomicpositions Ri in the crystal lattice. However, their implementations are highly dependent on the basisset used in the electronic structure code.

Among them, the “muffin-tin orbitals” in their linear version (LMTO) [5] form an adaptive mini-mal set of basis functions. They have each a well-defined momentum and are already well-localized.Most implementations have up to now used –almost directly – this basis set to span the correlatedsubspace C. Nevertheless, there are several possible choices of basis even within the LMTO-NMTOmethod. Basically, a compromise has to be made between the degree of localization and the orthogo-nality of the basis set. The most localized basis functions are not orthogonal and will therefore involvean overlap matrix. Since DMFT neglects non-local correlations, they may be the best one to choose.However, a non-orthogonal basis set may not be simple to implement for technical reasons1. On thecontrary, orthogonal LMTO-NMTO basis sets are somewhat more extended.

Recently, some approaches were developed to really construct Wannier functions. In 2000, Ander-sen et al. proposed the Nth order version of muffin-tin orbitals (NMTO) scheme [6] in which Wannier-like functions can be designed by using a “downfolding” procedure. Within such a Wannier basis,LDA+DMFT has been implemented and successfully applied to investigate the Mott transition inorthorhombic 3d1 pervoskites [130].

The “maximally-localized Wannier functions” proposed by Marzari, Vanderbilt and Souza [111, 155]were used in a LDA+DMFT implementation in 2006 [101]. The alternative projection proceduredeveloped by Anisimov et al. [10] was also recently applied within the LDA+DMFT framework [1, 3].Since the study carried out on Sr2IrO4 and presented in this thesis was performed with the lattermethod, we will describe it more precisely in section 2.3.

1This is mostly related to the use of impurity solvers, like QMC methods.


2.1.4 Approximations in the LDA+DMFT method

The important success of the LDA+DMFT approach must not hide the limitations of the formalism.In order to put into perspective this formalism, we summarize here the approximations performed onvarious level of the theory and try to justify them as far as possible.

The choice of the “correlated orbitals”

Only a few number of orbitals, the “correlated ” ones, are treated with DMFT in contrast to the “weaklycorrelated ” other orbitals. This arbitrariness is justified by physical considerations: by looking at theLDA band structure, the s and p bands are much broader than the d bands. The former are thus lesscorrelated than the latter. The ultimate test, of course, would involve treating all the orbitals withinDMFT, but still remains impractical nowadays.

The DMFT approximation

By using DMFT to solve the original lattice model, one assumes that the self energy is local – ormomentum-independent.

∀iωn ∀i Σii(iωn) ≈ Σimp(iωn) and ∀i, j Σij(iωn) ≈ 0. (1.50)

As explained in section 1.4, this approximation becomes exact in the limit of infinite coordinationnumber. The usual justification is that the coordination number of the considered system – like aperovskite structure – is “large enough”. A more rigorous argument could be provided by the comparisonto cluster calculations which reintroduce some momentum dependence. However, these simulations areoften not within reach of our computer systems.

The choice of the interaction Hamiltonian HU

This point was not really discussed. In practice, one often chooses to work with pure density-densityinteractions. As a result, Coulomb and Ising-like Hund’s coupling terms are included, but the fullrotational symmetry in spin space is not conserved anymore. It is clear that this simplification mightchange the physics, and the approximation is not really controlled.

The choice of “double-counting correction” Σdc

The choice of the correction for the wrongly assumed weak correlation within the LDA is not at allclear. While some schemes have been proposed, they are neither unique nor thoroughly derivable. Theeffect of these terms is a major source of uncertainty.

The “one-shot” implementation of LDA+DMFT

The LDA+DMFT cycle summarized on figure 2.1 should be iterated until the charge density ρ(r), theimpurity self-energy Σimp(iωn) and the chemical potential µ are converged. As already explained, mostimplementations do not perform the self-consistency over the charge density. This introduces a newapproximation in the calculation, which is not really controlled. It is however commonly thought thatthe “one-shot” results are really close to those which would be obtained with a full self-consistentlyimplementation. Some developments in the field – allowed by the growing computer power – arecurrently in progress to determine the validity of this belief.


2.2 Introduction to (linearized) augmented planewaves ( (L)APW )

We have chosen to implement the LDA+DMFT formalism within the “ linearized augmented planewaves(LAPW) framework ”, since recent developments proved the accuracy of using this kind of basis formany materials. More precisely, our implementation is based on the electronic structure code Wien2k[23], which is an all-electron full-potential LAPW method.

In this section, we first introduce the main characteristics of this code and then describe thoroughlythe construction of the APW and LAPW bases. For the interested reader, additional details about the(L)APW bases and Wien2k can be found in the following extensive reviews and books [23, 40, 149].

2.2.1 Wien2k, an all-electron full-potential LAPW method

In this part, we introduce the main concepts on which the program Wien2k relies. The interestedreader may find more details on the structure and the implementation of this code in the tutorial [23].Some additional technicalities are also described in Appendix C.

In Wien2k, the space contained in the unit cell of a given compound is partitioned into two regions:

• a set of non-overlapping spheres SαMT of radius RαMT around each atom α. They are called the“muffin tin spheres” and define the first region.

• the second region corresponds to the remaining space outside the spheres. It is called the “inter-stitial region” and labeled I.

This spatial division is linked to the different behaviors of the electronic wavefunctions in a crystal.Far away from the atomic nuclei, only the valence electrons can be found. In this region – whichcorresponds to I –, they are delocalized all over the solid and can be described by planewaves. Closeto the nuclei however, the core electrons, which do not participate significantly in chemical bondingwith other atoms, lie in atomic-like orbitals.

This partition in two regions of the unit cell was originally proposed by Slater in 1953 [152] andtwo different types of basis were then developed from this idea:

• the “ linear muffin-tin orbitals” (LMTO) [5] or its most recent Nth order version (NMTO) [6],

• the “augmented planewaves” (APW) and their descendants. These bases can be seen as madeof oscillating functions – like planewaves – that run through the unit cell. However, this simpleoscillating behavior is changed into something more complex inside the muffin-tin sphere of eachatom.

The Wien2k program is based on this last type of basis set, namely the “ linearized augmented planewave”(LAPW).

However not all the electronic wavefunctions are expanded within this basis set. Wien2k indeedintroduces the following difference:

• a “core state” is entirely contained in a muffin-tin sphere and is thus calculated by solving therelativistic radial Schrödinger equation for the considered free atom.

• a “valence state” requires on the contrary a description by (L)APW, since their wavefunctionsleak outside the muffin-tin spheres. A small subtlety is moreover considered here: the electroncan indeed be “pure valence” state or “semi-core” valence state. The latter are distinguished fromthe former because they lie high enough in energy so that their charge is not completely confined

2.2. INTRODUCTION TO (LINEARIZED) AUGMENTED PLANEWAVES ( (L)APW ) 31

inside the muffin-tin sphere, but they have only a few percent of it outside the sphere. Forinstance, in the case of Sr2IrO4, the orbitals Ir-5p and 4f are semi-core states. This distinctioninduces slight differences in the basis expansion, which is explained later.

Despite, Wien2k is an “all-electron” method: although core and valence electrons are not treatedin the same way, both of these states are indeed calculated self-consistently. The influence of the corestates on the valence is carried out by the inclusion of the core density ρcore(r) in the Hartree VH [ρ(r)]and exchange-correlation potentials V xc[ρ(r)] used to calculate the valence states, since:

ρ(r) = ρcore(r) + ρvalence(r). (2.18)

Reciprocally, the core states are calculated using the spherical average of the total electronic potentialρ(r) inside each muffin-tin sphere.

Moreover, the division of the unit cell into two domains induces also a dual representation for thedensity ρ(r) and the potential VKS(r). For instance, the Kohn-Sham potential VKS(r) is expanded asfollows:

VKS(r) =

∑

G

VGeiG·r if r ∈ I

Lmax∑

L=0

+L∑

M=−L

VLMYLM (rα) if r ∈ SαMT (α = 1, ..., Nat)

(2.19)

where G are the reciprocal lattice vectors and Y LM (r) the spherical harmonics. Since there is no shape

approximation for V (r), this method is also called “ full-potential ”. This represents the main advantageof the APW method to the LMTO-NMTO framework: within LMTO-NMTO, the decomposition ofthe potential corresponds to retaining only the L = 0 component inside the spheres and the G = 0component in the interstitial space. The recent introduction of the atomic sphere approximation (ASA)[4] has allowed to improve significantly the description of the potential but holds mainly for closelypacked structure. On the contrary, the APW method is free of any approximation and can then beconsidered as more accurate, especially for non-compact structures.

2.2.2 APW and LAPW bases

In Wien2k, the eigenfunctions of the Kohn-Sham Hamiltonian ψσkν(r) are expanded as follows:

ψσkν(r) =∑

|K|≤Kmax

cσKν(k)φkσK (r) (2.20)

with ν the band index, σ the spin degree of freedom and K the reciprocal lattice vectors. The basisfunctions φkσK (r) are of course the APW or LAPW functions and the parameter Kmax defines thecutoff of the basis expansion. We now describe thoroughly the construction of these APW and LAPWfunctions. For this, we follow the historical development of the method in order to be as clear aspossible, as it is done in [40, 103, 149].

The augmented planewaves (APW) basis

The notion of “augmented planewaves” (APW) was originally introduced in 1953 by Slater [152]. Inthis seminal work, the division of the unit cell is also exposed. However, the potential is taken asspherically symmetric inside the spheres and constant outside.

In order to describe at best the behavior of the electrons in each region, the basis functions φkσK (r)are expanded in planewaves in the interstitial region I and in atomic-like orbitals in the muffin-tin


spheres:

φkσK (r) =

1√Ωei(k+K)·r if r ∈ I

lmax∑

l

+l∑

m=−l

Aα,k+Klm uα,σl (rα, ε)Y l

m(rα) if r ∈ SαMT

(2.21)

where in this expression,

• K are the reciprocal lattice vectors, Ω is the unit cell volume, Nat is the total number of atomsin the cell, and ε has the dimension of an energy.

• rα = Tα(r) = Rα(r− rα) is the position inside the muffin-tin sphere SαMT , given with respect tothe center of the sphere rα. Tα is the transformation from the global coordinates of the crystalto the local coordinates associated to the atom α, which is composed of the rotation Rα and thetranslation by the vector rα. The length of rα is denoted rα, and the angles θα and φα specifyingthe direction of the vector in spherical coordinates, are indicated as rα.

• Y lm(r) are the spherical harmonics within the standard convention of Condon-Shortley (a factor

(−1)m is included in the definition).

• uα,σl (r, ε) is the radial solution of the Schrödinger equation for the free atom α at the energy ε:

[− d2

dr2+l(l + 1)

r2+ V (r)− ε

]ruα,σl (r, ε) = 0 (2.22)

However, the boundary condition is changed: for a true free atom, the boundary condition isthat uα,σl (r, ε) should vanish for r → +∞. In the APW method, it is required that the planewaveoutside the sphere matches the function inside the sphere over the complete surface of the sphere– in value, not in slope. If an eigenfunction was discontinuous, its kinetic energy would indeednot be well-defined. In addition, the following normalization condition is specified:

∫ RαMT

0r2|uα,σl (r, ε)|2dr = 1 ∀α, σ, l, ε (2.23)

• the Aα,klm are the expansion coefficients which are uniquely defined by requiring the continuity ofthe basis wavefunction at the sphere boundary and are given by:

Aα,k+Klm =

4π√Ωil Y m∗

l (Rα(k+K))jl(‖k+K‖RαMT )

uα,σl (RαMT , ε)ei(k+K)·rα (2.24)

where jl(x) is the Bessel function of order l.

To obtain this expression, the expansion of the planewaves in spherical harmonics around the originrα of the sphere of atom α were used:

1√ΩeiK·r =

4π√Ω

+∞∑

l=0

+l∑

m=−l

il jl(‖K‖‖r‖) Y m∗l (Rα(K) Y m

l (rα) (2.25)

In expression (2.25), the sum on the right-hand side is infinite. In practice, however, this sum must betruncated at some value lmax.

The APW method, as it is presented here, is not used anymore because of a major drawback.Indeed, to describe accurately an eigenstate ψσkν with the APW basis set, one has to solve inside the


muffin-tin sphere the radial Schrödinger equation at the band energy ε = εσkν . Consequently, the finaleigenvalue equation becomes non-linear and its solution much more computationally demanding foreach k-point. Another shortcoming of the APW method, known as the asymptote problem, is relatedto the indetermination of the expansion coefficients when the radial function uα,σl has a node at themuffin-tin radius RαMT . In the vicinity of this region, the relation between Aα,k+K

lm and cσKν(k) becomesnumerically unstable since uα,σl (RαMT , ε) is actually in the denominator of (2.24).

The regular LAPW method

In 1975, Andersen proposed a modification of the APW method [5] with the aim of overcoming itslimitations. In this approach, both the wavefunctions and their derivatives are made continuous atthe muffin-tin radius by matching the interstitial planewaves to a linear combination of the radialfunctions, and their energy derivative, calculated at a fixed reference energy.

By making a Taylor expansion of the radial wavefunction uα,σl around a reference energy ε0, onegets:

uα,σl (rα, ε) = uα,σl (rα, ε0) + (ε− ε0)uα,σl (rα, ε0) +O

((ε− ε0)

2)

(2.26)

with uα,σl (rα, ε0) =∂uα,σl (rα, ε)

∂ε

∣∣∣∣∣ε=ε0

In this expression, it is advantageous to choose ε0 near the center of the considered band. As a resultone should not choose one universal value ε0, but a set of well-chosen Eα1l up to l = lmax.

The definition of the LAPW basis set is then given by substituting the first two terms of theexpansion in the APW description (2.21), for a fixed Eα1l for each atom α and each value of l:

φkσK (r) =


lmax∑

l

+l∑

m=−l

[Aα,k+Klm uα,σl (rα, Eα1l) +Bα,k+K

lm uα,σl (rα, Eα1l)]Y lm(r

α) if r ∈ SαMT

(2.27)

A yet undetermined coefficient Bα,k+Klm is introduced because of the linearization. The expansion

coefficients Aα,k+Klm and Bα,k+K

lm are then obtained by requiring that φkσK (r) is continuous in value andslope at the sphere boundary SαMT . This leads to:

Aα,k+Klm =

4π√Ωil Y m∗

l (Rα(k+K)) Rα 2MT aαl (k+K) ei(k+K)·rα (2.28)

Bα,k+Klm =

4π√Ωil Y m∗

l (Rα(k+K)) Rα 2MT bαl (k+K) ei(k+K)·rα (2.29)

where

aαl (k+K) = ‖k+K‖ j′l(‖k+K‖RαMT ) uα,σl (RαMT , E

α1l)− jl(‖k+K‖RαMT )

[uα,σl

]′(RαMT , E

α1l)

bαl (k+K) = jl(‖k+K‖RαMT )[uα,σl

]′(RαMT , E

α1l)− ‖k+K‖ j′l(‖k+K‖Rα) uα,σl (RαMT , E

α1l).

(2.30)

We used the notation [...]′ = ∂[...]/∂r in the previous formula and we remind that uα,σl and uα,σl areorthogonal because of the normalization condition (2.23).


Within this linearized treatment, the error in the wavefunction is of second order in (ε − Eα1l).Taking into account the variational principle, this leads to an error of fourth order, (εσkν − Eα1l)

4 ,in the band energy. In other words, the LAPW basis set forms a good basis over a relatively largeenergy region, typically allowing the calculation of all the valence bands with a single set of referenceenergies Eα1l. Inded, the method rapidly demonstrated its power and accuracy. It has even become thebenchmark for electronic structure calculations within the Kohn-Sham scheme for decades.

The LAPW with Local Orbitals (LO) basis

There are situations in which the use of a single set of reference energies Eα1l is inadequate for all thebands of interest. Such a situation arises when two (or more, but rarely) states with the same l numberare involved in a chemical bonding or when bands over an unusually large energy region are required.This is precisely the case for the so-called “semi-core” states. To describe at best such states, the localorbitals (LO) were introduced by D. J. Singh in 1991 [148]:

φLOlm,α(r) =

0 if r ∈ I[Aα,LOlm uα,σl (rα, Eα1l) +Bα,LO

lm uα,σl (rα, Eα1l) + Cα,LOlm uα,σl (rα, Eα2l)]Y lm(r

α) if r ∈ SαMT

(2.31)These orbitals are defined for a particular l and m, and for a particular atom α. Moreover, they

are zero in the interstitial region and in the muffin tin spheres of other atoms. That is why, they arecalled “ local orbitals”. Local orbitals are strictly speaking not connected to planewaves in the interstitialregion, they have thus no k or K-dependence.

The three expansion coefficients Aα,LOlm , Bα,LOlm and Cα,LOlm are determined by requiring the local

orbital and its radial derivative to be zero at the muffin-tin sphere boundary. Through a third criterion,the local orbital can be associated to a fictitious planewave – ei(k+KLO)·r for instance – so that theorbital will behave just like the augmented planewave under inversion symmetry [149, 150]. However,we stress that local orbitals only exist inside the related muffin-tin sphere.

Adding local orbitals of course increases the LAPW basis set size. This slightly increases thecomputational time but is the price to pay for the much better accuracy that local orbitals offer todescribe the semi-core states.

The “APW plus local orbitals” (APW+lo) basis

The prohibitive shortcoming of the APW method was the energy dependence of the basis set. Thisenergy dependence have been removed in the LAPW(+LO) method, by increasing the size of the basisdue to the linearization in energy and the introduction of local orbitals (LO). An alternative methodin which the APW basis set may become energy independent but still have (almost) the same size asin the original APW method, was proposed in 2000 by Sjöstedt et al. [150, 151]. These two propertieswere obtained at the cost of adding a set of “ local orbitals”: the basis was then called “APW plus localorbitals”(APW+lo).

Two different types of functions compose the complete APW+lo basis set:

• the original APW basis functions described by (2.21) which are defined at chosen fixed lineariza-tion energies (Eα1l):


φkK(r) =


lmax∑

l

+l∑

m=−l

Aα,k+Klm uα,σl (rα, Eα1l)Y

lm(r

α) if r ∈ SαMT

(2.32)

with Aα,k+Klm =

4π√Ωil Y m∗

l (Rα(k+K))jl(‖k+K‖RαMT )

uα,σl (RαMT , Eα1l)

ei(k+K)·rα

• the set of local orbitals (lo) to keep the flexibility of the basis set with respect to the referenceenergy:

φlolm,α(r) =

0 if r ∈ I[Aα,lolm uα,σl (rα, Eα1l) +Bα,lo

lm uα,σl (rα, Eα1l)]Y lm(r

α) if r ∈ SαMT

(2.33)

These orbitals use the same linearization energies, although this is not strictly needed. The expansioncoefficients Aα,lolm and Bα,lo

lm are calculated by imposing the function to be zero at the sphere boundaryand by associating the local orbital to a fictitious planewave, as previously mentioned for the LO.

As in the LAPW method, a second set of local orbitals, labeled LO, can be introduced in order totreat semicore states:

φLOlm,α(r) =

0 if r ∈ I[Aα,LOlm uα,σl (rα, Eα1l) + Cα,LOlm uα,σl (rα, Eα2l)

]Y lm(r

α) if r ∈ SαMT

(2.34)

In contrast to the LO for LAPW, there is no derivative of uα,σl (rα, Eα1l) in the above expression. Theexpansion coefficients Aα,LOlm and Cα,LOlm are determined by matching the function to zero at the sphereboundary and by associating the local orbital to a fictitious planewave.

2.2.3 The LAPW basis with local orbitals or (L)APW+lo basis in Wien2k

We have previously introduced two types of basis set, namely the LAPW+LO and the APW+lo. Nowa-days, the state-of-the-art method, like the Wien2k code, is a combination of these both approaches.The APW+lo basis is used for valence d and f states but also for states in atoms that have a muffin-tin sphere which is much smaller than the other spheres in the unit cell. For all the other states, theLAPW+LO basis is employed. This combination is known as the “(L)APW+lo method ”.

Whatever the choice for the basis, any eigenstate of the Kohn-Sham Hamiltonian ψσkν(r) can thusalways be written as follows in Wien2k:

ψσkν(r) =

1√Ω

∑

|K|≤Kmax

cσKν(k)ei(k+K)r if r ∈ I

lmax∑

l=0

+l∑

m=−l

[Aναlm(k, σ)u

α,σl (rα, Eα1l) +Bνα

lm(k, σ)uα,σl (rα, Eα1l)

+ Cναlm (k, σ)uα,σl (rα, Eα2l)]Y lm(r

α) if r ∈ SαMT

(2.35)

More precisely, inside the muffin-tin spheres, the coefficients Aναlm(k, σ), Bναlm(k, σ) and Cναlm (k, σ) are

given by:


• if the atom α is described in the LAPW(+LO) representation,

Aναlm(k, σ) =∑

|K|≤Kmax

cσKν(k)Aα,k+Klm +

LOMAX(α,l,m)∑

nLO=0

cν,σLOAα,LOlm

Bναlm(k, σ) =

∑

|K|≤Kmax

cσKν(k)Bα,k+Klm +

LOMAX(α,l,m)∑

nLO=0

cν,σLOBα,LOlm

Cναlm (k, σ) =

LOMAX(α,l,m)∑

nLO=0

cν,σLOCα,LOlm (2.36)

• if the atom α is described in the APW+lo representation,

Aναlm(k, σ) =∑

|K|≤Kmax

cσKν(k)Aα,k+Klm +

lomax∑

nlo=1

cν,σlo Aα,lolm +

LOMAX(α,l,m)∑

nLO=0

cν,σLOAα,LOlm

Bναlm(k, σ) =

lomax∑

nlo=1

cν,σlo Bα,lolm

Cναlm (k, σ) =

LOMAX(α,l,m)∑

nLO=0

cν,σLOCα,LOlm (2.37)

where LOMAX(α, l,m) can be 0, if no LO are needed, or the number of necessary LO.

The reader has certanily noticed that all the previous orbitals were defined in the unit cell. Toextend their definition to the total crystal, it is enough to perform the following operation:

φµ(r) =1√N∑

R∈B

eik·Rφµ(r−R) with φµ = φkK, φlolm,α or φLOlm,α (2.38)

where N is the number of unit cells in the volume of the crystal and R the translation vectors of theBravais lattice. With this definition, the basis functions in the interstitial part are now ei(k+K)·r/

√V ,

with r running in all the crystal volume V .

2.3 Projection onto Wannier orbitals

The implementation of LDA+DMFT [1] which has been extended to perform our calculation on Sr2IrO4

relies on the construction of Wannier functions from the previously described APW+lo basis. In thissection, we first define more precisely the concept of “Wannier functions” which was briefly introducedin section 1.3. We then explain how the Wannier-function formalism based on the projection proceduredeveloped by Anisimov et al. [10] was applied within the APW+lo framework.

2.3.1 Wannier functions: definition and calculations

As explained in section 1.3, Wannier functions are the Fourier transformation of the Bloch states:

χRνσ(r) =

1√N∑

k

e−ik.Rψσkν(r) (2.39)

2.3. PROJECTION ONTO WANNIER ORBITALS 37

where N is the number of k-point in 1BZ (or the number of unit cells in the crystal), R a translationvector of the Bravais lattice, ν the band index2 and σ the spin degree of freedom. The concept was firstintroduced in 1937 by Wannier [163], in order to get a basis set whose functions are centered on theatomic positions in the crystal lattice. However, in most cases – and in contrast to the simple exampledescribed in section 1.3 –, Wannier functions are not uniquely defined: if one considers a certain setof bands ν, any orthogonal linear combination of Bloch functions |ψkν > can indeed be used in thedefinition (2.39):

χRµσ(r) =

1√N∑

k

e−ik.R∑

ν

U (k)µν ψ

σkν(r) (2.40)

where Ukµν is a unitary transformation matrix. To calculate in practice Wannier functions, one must

thus introduce an additional restriction on the properties of Wannier functions, in order to fix thisdegree of freedom.

First methods

The first methods which were proposed were based on an iterative optimization of trial functions whichhave the same real and point group symmetry properties as the desired Wannier functions. Amongthese methods, one can cite the variational Koster-Parzen principle [95, 129] which was later generalizedby Kohn [89, 92, 90, 91, 137], the general pseudopotential formalism proposed by Anderson [8] and theprojection operator formalism introduced by Cloizeaux [45, 46, 47]. However, all these computationalschemes were restricted to simple band structures.

The condition of “maximum localization”

In 1997, Marzari, Vanderbilt and Souza proposed the condition of “maximum localization” to calculateWannier functions [111, 155]. In order to ensure a maximally localized Wannier-like basis, the unitarymatrix U

(k)µν is obtained from a minimization of the sum Ω of the quadratic spreads of the Wannier

probability distributions defined by:

Ω =∑

µ,σ

(〈r2〉µσ − 〈r〉2µσ

)where 〈O〉µσ =

∫d3rO(r)|χ0

µσ(r)|2 with O = r or r2. (2.41)

Therefore, the quantity Ω may be understood as a functional of the Wannier basis set. Starting fromsome initial guess for the Wannier basis, the formalism uses steepest-descent or conjugate-gradientmethods to optimize U (k

µν . The resulting maximally-localized Wannier functions turn out to be realfunctions, although there is no available general proof for this property.

The projection procedure

A simpler alternative to this implementation was proposed in 2005 by Anisimov et al. [10]. The methodroughly consists in using atomic orbitals which are promoted to Wannier functions by a truncated ex-pansion over Bloch functions followed by an orthonormalization procedure.

More precisely, one needs first to define the site-centered atomic-like trial orbitals φµσ(r) in theunit cell and extend their definition to the total crystal by calculating:

φkµσ(r) =1√N∑

R∈B

eik·Rφµσ(r−R) (2.42)

2In order to keep our definition as general as possible, the Wannier functions χRνσ(r) will be now labelled directly by

the band index ν, without any reference to a corresponding atomic character.


One must then find the set W of corresponding physically relevant Bloch states |ψσkν〉 in the electronicband structure. These states – or bands – can be identified either by their band indices ν or bydelimiting the energy interval [ε1, ε2] in which they are located. The Wannier functions in the reciprocalspace χk

µσ(r) are then obtained by performing the following projection:

|χkµσ〉 =

∑

|ψσkν〉∈W

|ψσkν〉〈ψσkν |φkµσ〉 (2.43)

where the sum runs over the band indices ν only. A standard orthonormalization procedure is thenperformed, since in general only a subset of Bloch states are used in the previous expression.

With this definition, the obtained Wannier functions χkµσ(r) are not unique: they depend on the

energy window [ε1, ε2] covered by the Bloch functions: the larger this window, the more localizedthe Wannier functions. Despite this drawback, this procedure gives a quite straightforward scheme toconstruct Wannier functions. As a result, it was recently integrated in some LDA+DMFT implementa-tions based on the projector augmented wave (PAW) and the mixed-basis pseudopotential framework[3] but also on the (L)APW+lo framework of Wien2k [1].

2.3.2 Projectors on Wannier functions within the (L)APW+lo basis of Wien2k

As explained in section 2.1, the main physical issue in LDA+DMFT is to construct the localizedbasis set |χαL〉 which will span the correlated subspace C. However, these basis functions are only

involved through the projection operators P(C)α in the equations (2.4), (2.6) and (2.8) of the DMFT

self-consistent loop.Therefore, the program interfacing the DFT code and the DMFT part must merelydetermine the matrix elements of these projectors P

(C)α in the suitable bases.

This step is performed by the program “dmftproj” in the implementation we extended to performour LDA+DMFT calculations on Sr2IrO4. We describe here the main structure of dmftproj as it wasinitially developed by M. Aichhorn, L. Pourovskii and V. Vildosola [1]. The more general implemen-tation – in which the spin-orbit coupling may be taken into account – will be introduced in section3.3.

Definition of the “Wannier projectors” Pα,σlm,ν(k)

As already mentionned in section 2.1, the DMFT self-consistent loop involves two different bases:

• the Bloch basis |ψσkν〉, which is used to describe the lattice quantities,

• the Wannier functions |χα,σlm 〉 which define a localized basis set, where α specifies an atom in theunit cell, (l,m) are the orbital indices and σ the spin degree of freedom.

We define the “Wannier projectors” Pα,σlm,ν(k) as the matrix elements of the projectors P(C)α between

these two bases. They have the following expression:

Pα,σlm,ν(k) = 〈χα,σlm |ψσkν〉 (2.44)

and are explicitly calculated in the program dmftproj. With this definition, it is possible to specifythe formulas (2.4), (2.6) and (2.8) which give respectively the expression of the initial dynamicalmean-field G0:

[G0(iωn)]σmm′ =

∑

k,νν′

Pα,σlm,ν(k) [GKS(k, iωn)]σνν′

[Pα,σlm′,ν′(k)

]∗(2.45)

2.3. PROJECTION ONTO WANNIER ORBITALS 39

of the lattice self-energy correction Σσνν′(k, iωn):

Σσνν′(k, iωn) =∑

α,mm′

[Pα,σlm,ν(k)

]∗[∆Σαimp(iωn)]

σmm′ P

α,σlm′,ν′(k) (2.46)

and of the local Green’s function Gαloc(iωn):

[Gαloc(iωn)]

σmm′ =

∑

k,νν′

Pα,σlm,ν(k) Gσνν′(k, iωn)


]∗. (2.47)

Construction of the Wannier projectors within the (L)APW+lo basis

The calculation of the Wannier projectors in “dmftproj” follows the procedure introduced by Anisimovet al. [10]. For the interested reader, a brief tutorial to the program is provided in Appendix D.

If the orbital (l,m) of the atom α is considered as correlated, the solution of the Schrödingerequation within the corresponding muffin-tin sphere SαMT at the corresponding linearization energyEα1l is used as the trial orbital:

|φα,σlm 〉 = |uα,σl (Eα1l)Ylm〉 (2.48)

Independently of this choice, the user must also choose himself the suitable energy window W = [ε1, ε2]in which lie the Bloch states |ψσ

kν〉 used to perform the projection. From these definitions, the Wannier-like functions |χα,σlm 〉 are given by:

|χα,σlm 〉 =∑

|ψσkν〉∈W

|ψσkν〉〈ψσkν |uα,σl (Eα1l)Ylm〉 (2.49)

where the sum runs both on the band indices ν and on the k-points of 1BZ. The correspondingtemporary projectors are:

Pα,σlm,ν(k) = 〈uα,σl (Eα1l)Ylm|ψσkν〉 ∀ν ∈ W ∀k ∈ 1BZ. (2.50)

Since an energy window was chosen, the number of bands in the sum of (2.49) depends on k andσ. In practice, we define the temporary projection matrix Pα,σmν (k) for all the bands ν such thatνmin(k, σ) ≤ ν ≤ νmax(k, σ).

Using the decomposition of the Bloch states on the (L)APW+lo basis given by the general equation(2.35) and the following relations – which are deduced from the definition of the (L)APW+lo basis –:

〈uα,σl (Eα1l)Ylm|uα

′,σ′

l′ (Eα′

1l′)Yl′

m′〉 = δαα′δll′mm′δσσ′ ∀Eα1l, Eα′

1l′

〈uα,σl (Eα1l)Ylm|uα,σl′ (Eα1l)Y

l′

m′〉 = 0 ∀Eα1l

〈uα,σl (Eα1l)Ylm|uα

′,σ′

l′ (Eα′

2l′)Yl′

m′〉 = Oα,σlm,l′m′δαα′δσσ′ 6= 0 ∀Eα1l, Eα

′

2l′

(2.51)

the temporary projectors can be merely calculated by:

Pα,σlm,ν(k) = Aναlm(k, σ) +

LOMAX∑

nLO=1

cν,σLOCα,LOlm Oα,σ

lm,l′m′ (2.52)

Due to the truncation in the sum over the Bloch states, the Wannier-like orbitals |χα,σlm 〉 should beorthonormalized in order to give the final set of Wannier functions. One thus needs to calculate the


overlap matrix:

〈χα,σlm |χα′,σ′

lm′ 〉 =∑

k∈1BZ

Oα,α′

m,m′(k, σ)δσσ′ (2.53)

with Oα,α′

m,m′(k, σ) =

νmax(k,σ)∑

ν=νmin(k,σ)

Pα,σlm,ν(k)[Pα

′,σlm′,ν(k)

]∗. (2.54)

However, for practical reasons, it is easier to remain in the reciprocal space to perform this orthonor-malization. Indeed, in this case, it is enough to multiply each temporary projector by the inverse squareroot of Oα,α

′

m,m′(k, σ) and the "true" Wannier projectors Pα,σlm,ν(k) finally reads:

Pα,σlm,ν(k) =∑

α′m′

[O(k, σ)]−1/2

α,α′

m,m′Pα

′,σlm′,ν(k) ∀ν ∈ W ∀k ∈ 1BZ. (2.55)

As a result, the program dmftproj calculates roughly the Wannier projectors Pα,σlm,ν(k) in two steps: a

first subroutine produces the temporary projectors Pα,σlm,ν(k) using the equation (2.52) and a secondsubroutine performs the orthonormalization described in (2.55).

From this short description, it appears that the Wannier projectors Pα,σlm,ν(k) are defined in thecomplex spherical harmonics basis. This does not imply that the program can only calculate theWannier functions in this basis. It is possible to get the projectors to the correlated orbitals in cubicsymmetry or in any desired basis |ϕ(l)

i 〉 by requiring it in the input file. For this, the program will read

the corresponding unitary transformation U (l)i,m = 〈ϕ(l)

i |Ylm〉, which will be then used to transform theprojectors:

Pα,σli,ν (k) =∑

lm

U(l)i,m Pα,σlm,ν(k) (2.56)

The Brillouin zone integration

As observed (2.47), a sum over the k-points in the first Brillouin-zone (1BZ) is necessary. However,in order to reduce the computational time, Wien2k code solves the Kohn-Sham equations for the k-points in the irreducible Brillouin zone (IBZ). In order to evaluate Gα

loc(iωn), it is thus necessary tocalculate first the unsymmetrized Green function in IBZ and then apply the symmetry operation S ofthe crystallographic space group G as follows:

[Gαloc(iωn)]

σmm′ =

∑

k∈1BZ

Gα,σmm′(k, iωn) =∑

S∈G

∑

nn′

Dl(S)mn[ ∑

k′∈IBZ

Gα,σnn′(k′, iωn) ω(k

′)

]Dl(S−1)n′m′

with Gα,σmm′(k, iωn) =∑

νν′

Pα,σmν (k) Gσνν′(k, iωn)

[Pα,σν′m′(k)

]∗ (2.57)

In this expression, k′ samples a tetrahedral mesh and ω(k′) represents its corresponding weight. Inprinciple, the tetrahedral weights ων(k′) depends also on the band index: its value is smaller for thebands which cross the Fermi level and it is 0 if the band is empty. In practice however, we take merelythe simple geometrical factor of the tetrahedron ω(k′) as corresponding weight, since in the DMFTcycle the chemical potential µ will change.

A similar approach is also used to calculate the initial dynamical mean-field G0 since a sum overthe k-points appears also in (2.45). More generally, any local quantity – density matrix, spectralfunction,. . . – require such a treatment to be evaluated. For the interested reader, the formula (2.57)is explicitly derived in Appendix E.

Chapter 3

Taking into account the spin-orbitinteraction in LDA+DMFT

In solid state physics, the spin-orbit interaction is commonly thought as a weak relativistic correc-tion to the Schrödinger equation and is thus treated perturbatively. However, its effects on the bandstructure of solids can be quite dramatic, quantitatively and even qualitatively. This was originallyhighlighted in semiconductors with the well-known “Dresselhaus splitting” [49, 50] and more recentlywith the discovery of “topological bands insulators” [21, 94, 118]. Furthermore, by coupling the spindegree of freedom with the electronic angular momentum, the spin-orbit interaction seems to fulfill theexpectations of “spintronics”: to control the spin orientation in solid with an electric field.

The interplay between electronic correlations and the spin-orbit coupling is a very new field ofinterest in condensed matter physics [131, 164]. Recent works – on strontium rhodate [64, 107] oriridium-based transition metal oxides [84, 116] to only name a few – have also shown the significantrearrangements in the band structure which can arise by taking into account these both interactions.In this context, developing an LDA+DMFT implementation for which the spin-orbit corrections canbe fully integrated in the definition of the correlated orbitals is of great interest.

In this section, the derivation of the spin-orbit interaction and its influence on an atomic level arefirst reminded. We then give a brief review of the effects induced by the spin-orbit coupling in solid statephysics. We finally describe how this relativistic correction may be introduced in the LDA+DMFTimplementation of Aichhorn et al. [1]. This improvement of the method is one of the major technicaldevelopments achieved during this thesis in order to perform the LDA+DMFT study of Sr2IrO4.

3.1 Basics on the spin-orbit interaction

In this section, we first derive the spin-orbit correction from the Dirac equation and then present theinfluence of this term at an atomic level by introducing the notion of “fine structure multiplets”. Atten-tion is particularly paid on the case of atomic d orbitals in cubic symmetry because of its importancein the case of Sr2IrO4. All these fundamental concepts on the spin-orbit interaction can be found inany reference book, but we specially recommend [154, 157].

3.1.1 Derivation of the spin-orbit coupling term

The “spin-orbit interaction” is a correction to the Schrödinger-Pauli equation in the limit where rel-ativistic effects are assumed weak. It introduces a coupling between the spin s and the motion – ormore precisely the orbital momentum l in the atomic case – of the electron.

41

42 CHAPTER 3. LDA+DMFT WITH SPIN-ORBIT COUPLING

Non-relativistic limit of Dirac’s Hamiltonian

Dirac’s Hamiltonian provides a description of an elementary spin-1/2 particles, such as electrons,consistent with both the principles of quantum mechanics and the theory of special relativity. Anelectron in an external potential V (r) is thus described by the following equation:

HDirac Ψ = i~∂Ψ

∂twith HDirac = c α · p+ βm0c

2 + V (r) (3.1)

where α and β are the 4× 4 matrices:

α =

(0 σ

σ 0

)and β =

(Id 00 −Id

)(3.2)

and σx ,σy and σz are the Pauli-spin matrices. The stationary solutions of (3.1) are the four-componentfunctions Ψ which can be written in terms of two 2-spinors Φ and χ as follows:

Ψ = e−iEt(

Φχ

)and

c [σ · p] χ = [E − V (r)−m0c

2] Φc [σ · p] Φ = [E − V (r) +m0c

2] χ(3.3)

For electrons – which are positive energy solutions of (3.1) –, Φ describes the “ large component” of thewave-function and χ the “small ” one.

The “spin-orbit coupling term” appears when one develops the Hamiltonian (3.1) in the “non-relativistic limit” up to the order (v/c)2. By shifting the energies reference by the rest energy m0c

2

of the electron (E = ε + m0c2), the coupled equations (3.3) determine the order of magnitude of χ

as (v/c) times smaller than Φ. As a result, the initial four-component problem can be reduced in thenon-relativistic limit to the following equation where only the large component Φ appears:

1

2m0[σ · p]

[1 +

ε− V (r)

2m0c2

]−1

[σ · p] Φ + V (r) Φ = ε Φ. (3.4)

The expansion of the denominator of the first term gives:[1 +

ε− V (r)

2m0c2

]−1

= 1− ε− V (r)

2m0c2+O

(1

m20c

4

). (3.5)

By using then the following operator identities:

p V (r) = V (r)p− i~∇V (r)

[σ ·∇V (r]) [σ · p] = ∇V (r) · p+ iσ · [∇V (r)× p] (3.6)

the differential equation (3.4) becomes:[(

1− ε− V (r)

2m0c2

)p2

2m0+ V (r)

]Φ − ~2

4m20c

2[∇V (r) ·∇Φ] +

~

4m20c

2σ · [∇V (r)× p] Φ = ε Φ

[p2

2m0+ V (r)

]Φ − p4

8m30c

2Φ − ~2

4m20c

2[∇V (r) ·∇Φ] +

~

4m20c

2σ · [∇V (r)× p] Φ = ε Φ.

(3.7)The first and the second term give the usual non-relativistic Schrödinger equation. The third and thefourth term are the mass and Darwin correction respectively. Finally, the last term corresponds to thespin-orbit coupling in its most general form:

HSO =~

4m20c

2σ · [∇V (r)× p]. (3.8)

3.1. BASICS ON THE SPIN-ORBIT INTERACTION 43

If the potential has the spherical symmetry – as, for instance, for a simple atomic nucleus –, we indeedobtain the more common expression:

HSO =1

2m20c

2

1

r

dV

dr(l · s) with s =

1

2~σ and l = r× p. (3.9)

“Intuitive” derivation in the atomic case

The spin-orbit interaction term HSO can also be derived in the framework of classical electrodynamicsif one considers a simple atom. Since there exits a magnetic moment of the electron, connected withthe electron spin s, this moment µ = −2µ0s/~, with µ0 = e~/2m0 the Bohr magneton, leads to anadditional interaction −µ ·B between the electron and the nucleus.

In this expression, B is the magnetic field which is associated with the electron moving in theelectric field E induced by the nucleus. Since B = −(v × E)/c2 in classical electromagnetism, theadditional energy of the electron in the field can be rewritten as:

−µ ·B = +e

m0s ·B = +

e

m20c

2s · (E× p). (3.10)

This term in the Hamiltonian is essentially the spin-orbit term HSO, except for a factor 2. This factor– the Thomas precession factor – is missing because the complete Lorentz transformation was notapplied. Actually, while changing the frame of reference, a time transformation also occurs and conse-quently, the precession frequency of the electron spin in the magnetic field is modified by a factor 1

2 .

Since the electrical field induced by the nucleus is central symmetric, one can write:

E = −∇

(−1

eV (r)

)=

1

e

∂V

∂r

r

r(3.11)

where V (r) is the potential of the atomic nucleus. As a result, the previous energy term (3.10) becomes:

+

(1

2

)e

m20c

2s · (E× p) =

e

2m20c

2s ·(1

e

∂V

∂r

r

r× p

)=

1

2m20c

2

1

r

∂V

∂r(l · s) (3.12)

which is exactly the spin-orbit coupling term (3.9).

3.1.2 Effects on atomic orbitals

According to the expression (3.9), the spin-orbit interaction depends on the value of the the angularmomentum l of the electron – the spin has the same value s = 1/2 for all electrons – and the mutualorientation of the angular momentum l and the spin s. In other words, the spin-orbit interactiondepends on the value of the total angular momentum j = l+ s:

HSO =1

2m20c

2

(1

r

∂V

∂r

)1

2(j2 − l2 − s2). (3.13)

Energy splitting for an hydrogen-like ion

Because of the spin-orbit coupling, the energy of an electron in the states j = l + 12 and j = l − 1

2 isnow different. To evaluate this energy splitting, one can calculate the mean value of the perturbationHSO in the atomic state (n, l) for an hydrogen-like ion:

〈HSO〉n,l =Ze2

4πε0

~2

2m20c

2〈 1r3

〉 1

2

[j(j + 1)− l(l + 1)− 3

4

]since V (r) = − Ze2

4πε0

1

r. (3.14)


Consequently, the correction to the energy due to the spin-orbit coupling is:

(∆ESO)n,l = ζSO1

2

[j(j + 1)− l(l + 1)− 3

4

](3.15)

with ζSO =e~2

8πε0m0c2a30

1

l(l + 1)(l + 12)

Z4

n3eV =

α2

l(l + 1)(l + 12)

Z4

n3Ry. (3.16)

To obtain this expression, we have used1 the following formula:

〈 1r3

〉 =∫

1

r3R2nl(r) r

2dr =1

n3l(l + 1)(l + 12)

Z3

a30. (3.18)

As a result, the spin-orbit corrections scale as Z4 where Z is the atomic number. Heavy elements willthus undergo more important spin-orbit corrections than light ones, as shown in table 3.1.

Atom Z ζSO(3d)

Iron (Fe) 26 0.050 eVCopper (Cu) 29 0.103 eV

Atom Z ζSO(4d)

Ruthenium (Ru) 44 0.161 eVRhodium (Rh) 45 0.191 eV

Atom Z ζSO(5d)

Iridium (Ir) 77 0.4 eVGold (Au) 79 0.42 eV

Atom Z ζSO(6p)

Bismuth (Bi) 83 1.5 eV

Table 3.1: Typical value of the spin-orbit constant ζSO for some elements of the periodic table. From[53] (5d), from [64] (4d), from Landolt-Börnstein database (3d) and from [71] (Bi)

Fine structure multiplets in multi-electronic atoms

Without spin-orbit interaction, the multi-electronic configuration of an atom is completely describedby its total angular momentum L and its total spin S, which form the LS spectral term of the system.The (2L + 1)(2S + 1) states – which differ by the value of the z component of the orbital and spinmomenta ML and MS – have all the same energy. The spin-orbit interaction leads to a splitting of thestandard LS spectral term into a number of components corresponding to different values of the totalangular momentum J of the atom. This splitting is called “fine” or “multiplet splitting”.

Multiplet splitting obeys a rule which is called “Landé’s interval rule”. According to this rule, thesplitting of the levels J and J − 1 is proportional to J :

∆EJ,J−1 = χ(LS).J (3.19)

The multiplet splitting constant χ(LS) is different for different spectral terms and can be of eithersign:

• When χ(LS) > 0, the multiplet component with the smallest possible value J = |L− S| has thelowest energy value. Such a multiplet is called “normal ”.

• When χ(LS) < 0, the multiplet component with the greatest possible value J = L + S has thelowest energy value. Such a multiplet is called “ inverted ”.

1We remind that the Bohr radius a0, the fine structure constant α and a Rydberg are defined by:

a0 =4πε0~

2

m0e2, α =

e2

4πε0~cand 1 Ry =

~2

2m0a20. (3.17)

3.1. BASICS ON THE SPIN-ORBIT INTERACTION 45

It has been empirically established that a configuration containing n equivalent electrons:

• corresponds to normal multiplets, when n < 2l + 1 (when shells are less than half-filled),

• corresponds to inverted multiplets, when n > 2l + 1 (when shells are more than half-filled),

• has no triplet splitting if n = 2l + 1.

Each spectral term, except for singlet terms and S terms, has a fine structure. In general however,the distance between the components of this structure is considerably less than the distance betweendifferent spectral terms. This grouping of levels is characteristic of the approximation which is calledthe “Russel-Saunders coupling” approximation. Since this is the most current case, the expression “LScoupling” or “normal coupling” is also used.

Analysis of experimental data have shown that the range of applicability of the LS coupling ap-proximation is actually limited. It is therefore of interest to consider another limiting case: whenthe spin-orbit interaction considerably exceeds the electrostatic interaction. This case is called “jj-coupling”. If the spin-orbit coupling is large, one can only speak of the total angular momentum of anelectron j, as only this angular momentum is conserved. jj coupling is rarely found in pure form inatomic spectra. However, the structure of the spectra of the heavy elements very closely approachesthe structure characteristic of jj-coupling. Generally speaking, in passing from the light to the heavyelements, a more or less continuous transition from LS to jj coupling is observed.

3.1.3 Atomic d orbitals in cubic symmetry and spin-orbit coupling

In many transition metal oxides – such as those which crystallize in the perovskyte or K2NiF4-typestructure for instance –, the transition metal ion is at the center of an octahedron, typically made bysix oxygen atoms. In this case, the local crystal field lifts the degeneracy of the d orbitals, which arethen divided in:

• a three-fold group of states called “t2g” (dxy,dxz and dyz) lower in energy,

• a doublet labelled “eg” (dx2−y2 and d3z2−r2) higher in energy.2

If the transition metal ion is heavy enough, like an iridium atom or any other 5d element, thespin-orbit interaction must be taken into account. As a result, the symmetry of the system will belowered and fine multiplets will appear. However, due to the cubic crystal field, the fine structure ofthe d orbitals will not be decomposed in a six-fold j = 5/2 group and a quartet j = 3/2 as for a singleatom. Our aim is now to derive it in the framework of the “TP-equivalence approximation”.

The matrix elements of the orbital angular momentum l in a cubic system with a single electront2g or eg are the following:

lx =

0 0 i 0 0

0 0 0 −i√3 −i

−i 0 0 0 0

0 i√3 0 0 0

0 i 0 0 0

, ly =

0 0 0 i√3 −i

0 0 −i 0 00 i 0 0 0

−i√3 0 0 0 0

i 0 0 0 0

(3.20)

and lz =

0 i 0 0 0−i 0 0 0 00 0 0 0 2i

0 0 0 0 00 0 −2i 0 0

2For the interested reader, a brief reminder on the definition of atomic d states in an octahedral ligand field can befound in Appendix A.


where the matrices are given in the basis dxz, dyz, dxy, d3z2−r2 , dx2−y2. The calculation is straight-forward by using the explicit forms of the t2g and eg orbitals and the well-known relations:

lz|ϕlm〉 = ~m|ϕlm〉 and l±|ϕlm〉 = ~√l(l + 1)−m(m± 1)|ϕlm±1〉 with l± = lx ± ily. (3.21)

The matrices (3.20) are hermitian and their elements are purely imaginary (the cubic basis is real).However, the matrix elements of l in the eg subspace are zero. This means that the orbital angularmomentum is “completely quenched ” in the eg states. As a result, there is no first-order spin-orbitinteraction for these states.

On the contrary, in the t2g subspace, the orbital angular momentum is not quenched. In addition,by comparing the matrix elements in the t2g states with those in the p states of a free atom, it appearsthat:

l(t2g) = −l(p). (3.22)

This means that the expectation value of l2 = l2x + l2y + l2z in the t2g subspace is l(l+ 1) not with l = 2but rather with l = 1: the orbital angular momentum is thus “partially quenched ” in the t2g states.The relation (3.22) is called the “TP-equivalence”.

The TP-equivalence is only formal as seen from the fact that l(t2g) does not satisfy the commutationrelation which the angular momentum should satisfy. The non-diagonal matrix elements between egand t2g states were indeed neglected. Nevertheless, if the cubic-field splitting between the eg and t2gstates is large, neglecting these elements may be justified and the TP-equivalence can conveniently beused. In this case, the matrix of the spin-orbit interaction in the t2g subspace can be decomposed intwo submatrices:

0 −i ii 0 −1−i −1 0

.

ζSO2

and

0 i 1−i 0 i−i 1 0

.

ζSO2

(3.23)

in the bases dxz↑, dyz↑, dxy↓ and dxz↓, dyz↓, dxy↑ respectively, and with:

ζSO =1

2m20c

2

∫ +∞

0

1

r

∂V

∂rR2nd(r) r

2dr. (3.24)

Both these matrices can be partially diagonalized by using the following linear combination of dxz anddyz:

|t+〉 = − 1√2(|dyz〉+ i|dxz)〉 and |t−〉 =

1√2(|dyz〉 − i|dxz)〉. (3.25)

As a result, in the bases t+ ↑, t− ↑, dxy↓ and t+ ↓, t− ↓, dxy↑, the matrices (3.23) become:

−1 0 0

0 1 −√2

0 −√2 0

.

ζSO2

and

1 0 −√2

0 −1 0

−√2 0 0

.

ζSO2. (3.26)

By diagonalizing the last 2× 2 block, one finally obtains:

• a doublet of eigenstates associated to the eigenvalue εSOjeff=

12

= +ζSO:

|jeff =1

2,mj = −1

2〉 = 1√

3|dyz ↑〉 −

i√3|dxz ↑〉 −

1√3|dxy ↓〉

|jeff =1

2,mj = +

1

2〉 = 1√

3|dyz ↓〉+

i√3|dxz ↓〉+

1√3|dxy ↑〉

(3.27)

3.2. EFFECTS OF THE SPIN-ORBIT COUPLING IN SOLIDS 47

• a quartet of eigenstates with the same eigenvalue εSOjeff=

32

= −ζSO/2:

|jeff =3

2,mj = −3

2〉 = 1√

2|dyz ↓〉 −

i√2|dxz ↓〉

|jeff =3

2,mj = +

3

2〉 = − 1√

2|dyz ↑〉 −

i√2|dxz ↑〉

|jeff =3

2,mj = −1

2〉 = 1√

6|dyz ↑〉 −

i√6|dxz ↑〉+

√2

3|dxy ↓〉

|jeff =3

2,mj = +

1

2〉 = − 1√

6|dyz ↓〉 −

i√6|dxz ↓〉+

√2

3|dxy ↑〉

. (3.28)

In these expressions, the analogy with the p 12

and p 32

was used to label the states. Furthermore, theseparation between the two multiplets is given by Landé’s interval rule:

εSOjeff=

32

− εSOjeff=

12

= −3

2ζSO. (3.29)

As a result, by assuming that the cubic splitting between the eg and t2g orbitals is much largerthan the spin-orbit splitting, the fine structure of the d atomic orbitals is given by:

• the eg states on which the spin-orbit interaction is ineffective, because of their quenched angularmomentum,

• the doublet jeff = 1/2 (3.27) and the quartet of states jeff = 3/2 (3.28), the former being higherin energy than the latter.

So far, we have neglected the non-diagonal matrix elements of the spin-orbit interaction, the completecalculation of the multiplets without the TP-equivalence is performed in the appendix A. In addition, acalculation within the TP-equivalence approximation but including a small tetragonal field between thet2g states is also described in this appendix. Neverthelees, as we will see in section 5, this descriptionof the atomic d orbitals within the “TP-equivalence approximation” is enough to study the effect of thespin-orbit coupling in Sr2IrO4.

3.2 Effects of the spin-orbit coupling in solids

In this section, a (very) general overview of the effects induced by the spin-orbit coupling in solid statesphysics is presented. Since many specific – and often new – branches are involved, we are not able toprovide many details in each field. We present first the well-known effects induced by the spin-orbitcoupling in semiconductor structures before merely listing the state-of-the-art discoveries which involvethe spin-orbit interaction in modern condensed matter physics.

3.2.1 Dresselhaus and Rashba terms

In 1954, Elliot [51] and Dresselhaus et al. [50] emphasize that the spin-orbit interaction may haveimportant consequences for the one electron energy levels in bulk semi-conductors. Subsequently, spin-orbit coupling effects in a bulk zinc blende structure were discussed in two classic papers by Parmenter[128] and Dresselhaus [49]. Unlike the diamond structure of silicon (Si) and germanium (Ge), the zincblende structure does not have a center of inversion. As a result, a spin splitting of the electron andhole states occurs at non-zero k-point even for a zero magnetic-field. This spin splitting was interpretedas a consequence of the spin-orbit coupling, because otherwise, the spin degree of freedom of the Bloch


electrons would not know whether it was moving in an inversion symmetric diamond structure or aninversion-antisymmetric zinc blende structure.

To understand better this last statement, it is necessary to consider the action of the time reversaloperator Θ on the system defined by the Hamiltonian H:

H =p2

2m0+ V (r) +

~2

4m20c

2σ · [∇V (r)× p] (3.30)

where V (r) is the lattice potential and the third term the spin-orbit correctionin its most general form(3.8). Since this Hamiltonian commutes with Θ, the Bloch function ψσk(r) and ψ−σ∗

−k (r) are associatedto the same eigenvalue:

εσ(k) = ε−σ(−k) (3.31)

with k the momentum and σ the spin index (for the sake of simplicity, band indices were omitted).

If moreover the crystal has a center of inversion – that is to say, if the operation I : r → −r belongsto the crystallographic spacegroup – , the following relation also holds:

εσ(k) = εσ(−k) (3.32)

By combining the equations (3.31) and (3.32), it becomes then clear that if both time reversal symmetryΘ and spatial inversion symmetry I are present in the system, the band structure should satisfy to thecondition:

εσ(k) = ε−σ(k) (3.33)

In other words, each band will preserve its spin degeneracy, as illustrated in figure 3.1 (a). This isthe case for the diamond structure of silicon and germanium. On the contrary, if a center of inversionsymmetry is absent in the crystal, like in the zinc blende structure of the semiconductors InSb or GaAs,only the relation (3.31) remains and, as shown in figure 3.1 (b), the spin degeneracy is lifted by thespin-orbit term (3.8). This phenomenon is called the “Dresselhaus splitting”.

Figure 3.1: Schematic example of the “Dresselhaus splitting”. In panel (a), the solid has a center ofsymmetry and the level is doubly degenerate. In panel (b), the solid has no center of inversion and theDresselhaus splitting occurs.

Nevertheless, the spin degeneracy can not only be lifted because of a bulk inversion asymmetry ofthe underlying crystal structure but also because of a structure inversion asymmetry of the confiningpotential V (r). Particularly, the termination of the crystal by a surface destroys the inversion symmetryin the direction of the normal. In this case, by performing a Taylor expansion of the potential V (r):

V (r) = V0 + er · E +O(r2) (3.34)

3.2. EFFECTS OF THE SPIN-ORBIT COUPLING IN SOLIDS 49

the lowest order of the inversion asymmetry of the potential is characterized by an electric field E . Asa result, the corresponding spin-orbit correction can be rewritten as:

HRashba =αR~

σ · (k× E) (3.35)

which is the “Rashba Hamiltonian” [29]. This term can induce significant effects in quasi-two-dimensionalsemiconductor structures, such as quantum wells and heterostructures [166]. Furthermore, exploitingthis effect is at the root of “semiconductor spintronics”, as we will see in the following.

3.2.2 New domains involving the spin-orbit coupling

As already mentioned in the introduction, the spin-orbit interaction has recently regained interest incondensed matter physics, especially through three different fields, which are briefly presented here.

Spintronics

“Spintronics” is a multidisciplinary field whose central theme is the active manipulation of spin degreesof freedom in solid state physics. The discovery of the giant magneto-resistance (GMR) in 1988 byFert et al. [18] and Grünberg et al. [22] has marked the beginning of the development of this field.

Very recently, in the field of “semiconductor spintronics” a number of spin-electronic devices havebeen proposed, which explicitly make use of the Rashba effect (3.35), motivated by the proposal of a“spin field-effect transistor ” by Datta and Das [44]. The basic idea is the control of the spin orientationby using this spin splitting due to the spin-orbit coupling in the presence of a structure inversionasymmetric potential. More details can be found in the review [162] for instance.

Topological band insulators

The discovery of “topological band insulators” in theory [21, 54, 82, 119] and experiment [71, 94] hasopened a new field which is nowadays very active. In these remarkable materials, the strong spin-orbitinteraction allows a non-trivial topology of the electron bands, resulting in protected “helical edge” and“surface states” in two and three dimensional systems. As a result, a topological band insulator, likean ordinary insulator, has a bulk energy gap separating the highest occupied electronic band from thelowest empty band but its surface – or edge in two dimensions – exhibits gapless electronic states thatare protected by time reversal symmetry.

Many other interesting phenomena, including “quantum number fractionalization” and “magneto-electric effects” have been predicted to occur in these systems, and are currently the subject of agrowing experimental effort.

Interplay between electronic correlations and the spin-orbit coupling

The interplay between the spin-orbit coupling and the electronic Coulomb correlations has recently be-come the subject of intense research in condensed matter physics [131, 164]. For instance, recent workson strontium rhodate (Sr2RhO4) have indeed shown that the spin-orbit coupling can be reinforced bythe action of the Coulomb interaction [64, 86, 107], whereas the spin-orbit coupling is considered asthe driving force for the Mott insulating state in strontium iridate (Sr2IrO4) [84].

In this field, the attention is particularly focused on the properties of iridium-based transition metaloxides – such as Na4Ir3O8 [36, 127], Na2IrO3 [146], Pr2Ir2O7 [125] and of course the Ruddlesden-Popperserie Srn+1IrnO3n+1 [116] –, but other frustrated magnets [159] or Fe-spinel [37] are also studied a lot.In this context, developing an LDA+DMFT implementation for which the spin-orbit corrections can


be fully integrated in the definition of the correlated orbitals is of great interest. The subject of thisthesis is therefore directly related to this very recent field of research.

3.3 Implementation of the spin-orbit coupling in LDA+DMFT

Theoretically, the spin-orbit interaction can easily be included inside the LDA+DMFT formalism.Indeed, the equations introduced in section 2.1 can still be used if any spin-index σ is forgotten3.In practice however, developing such an implementation of LDA+DMFT, which could be called an“LDA+SO+DMFT implementation”, is highly dependent on how the spin-orbit coupling is alreadytaken into account at the LDA level.

Since our aim during this thesis was to develop such an implementation based on the LAPWframework – extending then the LDA+DMFT implementation of Aichhorn et al. [1] –, we describefirst in this section how the spin-orbit interaction is treated within the Wien2k code. The consequenceon the definition of the Wannier projectors and on DMFT equations is then presented.

3.3.1 How the spin-orbit interaction is included in Wien2k

Wien2k provides the possibility of performing both non-relativistic and relativistic calculations. Whenrunning relativistic calculations, relativity is included in a way which differs for core and valence states.The core states are assumed to be fully occupied and a fully relativistic treatment is performed. Onthe contrary, the valence orbitals are only treated within the “scalar relativistic approximation”, whichactually consists in neglecting the spin-orbit interaction. The spin-orbit corrections can however bereintroduced later, via a second variational approach.

The scalar relativistic approximation

To introduce the scalar relativistic approximation, we have to consider the Hamiltonian obtained in(3.7) and assume that the potential V (r) is spherically symmetric. We remind that Ψ is not an eigen-function of spin s or orbital moment l within this description. Actually, the good quantum numbersare the total angular momentum j, its projection jz and κ4.

Despite, the four-component function Ψ can be written as:

Ψ =

(Φχ

)=

(g(r)Yj,ljzif(r)Yj,ljz

)(3.36)

where g and f are the radial function and Yj,ljz is the r-independent eigenfunction of j2, jz, l2 ands2 = 3

4 formed by the combination of the Pauli spinors ϕ↑, ϕ↓5 with the spherical harmonics Y l

m. One

3This implies of course to include the spin degree of freedom in the band index ν for all the quantities related to theKohn-Sham basis and to consider that L stands now for the indices j, jz, κ.

4This last quantum number is akin to the ± sign in j = l ± 1/2 in the non-relativistic limit. It can be shown thatκ = ±(j + 1

2). More precisely, ~κ is the eigenvalue of the operator K such that:

K =

(l · σ + ~ 0

0 −l · σ − ~

).

5We remind that ϕ↑ =

(10

)and ϕ↓ =

(01

).

3.3. IMPLEMENTATION OF THE SPIN-ORBIT COUPLING (SO) IN LDA+DMFT 51

can then derive the following coupled system of equations for f and g:

~c

[df

dr+

1− κ

rf

]= −[ε− V (r)] g

~c

[dg

dr+

1 + κ

rg

]= [ε− V (r) + 2m0c

2] f.(3.37)

By eliminating f , one gets:

− ~2

2Mr2d

dr

(r2dg

dr

)+

[V (r) +

~2

2M

l(l + 1)

r2

]g − ~2

4M2c2dV

dr

dg

dr− ~2

4M2c2dV

dr

1 + κ

rg = ε g (3.38)

where we have used the “relativistically enhanced mass” M = m0 + (ε− V (r))/2c2 and the relationκ(κ+ 1) = l(l + 1). The function f is on the contrary given by

f =~

2Mc

(dg

dr+

1 + κ

rg

). (3.39)

The scalar relativistic approximation is obtained by omitting, in the previous equations (3.38) and(3.39), the terms which depend on κ. Clear advantage of this approximation is that l and s are nowgood quantum numbers.

Let Ψ, Φ, χ, f and g be the scalar relativistic approximation of Ψ, Φ, χ, f and g. On the one hand,we have:

Ψ =

(Φχ

)with Φ = g Y l

m χs ∀ s ∈ ↑, ↓. (3.40)

since Φ is a pure spin-state. On the contrary, χ contains a mixture of up and down spin functionsbecause it is obtained from equation (3.3):

χ = iσ · rr

[−f +

g

2Mcrσ · l

]Y lm χs. (3.41)

On the other hand, the radial functions satisfy the following equations:

f =~

2Mc

dg

drand g = − ~c

ε− V (r)

df

dr(3.42)

which lead to the following equation for g:

− ~2

2Mr2d

dr(r2

dg

dr) + [V (r) +

~2

2Mr2l(l + 1)

r2] g − ~2

4M2c2dV

dr

dg

dr= ε g. (3.43)

From these definitions, the spin-orbit Hamiltonian HSO is obtained by the following relation:

HDiracP si = εΨ +HSOΨ. (3.44)

It can then be shown that in the scalar relativistic basis of function f and g:

HSO =~

2Mc21

r

dV

dr

(σ · l 00 0

)(3.45)

HSO acts only on the large component of the wavefunction and can then be included perturbatively.


Second variational treatment of the spin-orbit interaction in Wien2k

In all the previously derived equations, the quantity V (r) stands only for an external potential. How-ever, equations (3.42) and (3.43) are solved in Wien2k to construct the (L)APW+lo basis set by usingthe effective Kohn-Sham potential VKS(r) which is defined in (1.18). The many-body effects inducedby the Coulomb repulsion is thus included in this approach via the Hartree potential VH(r) and theexchange-correlation potential V xc(r). However, this does not imply that the initial many-body prob-lem is rigorously treated within a scalar relativistic approximation. As explained in Appendix F,many-body spin-orbit interaction terms, called “spin-same-orbit” and “spin-other-orbit” interactionsshould also be taken into account when the scalar relativistic approximation is derived from the Dirac-Coulomb-Breit Hamiltonian [26]. Nevertheless, we will not discuss here the effects of these additionalterms – the interested reader is invited to read Appendix F – and we will only focus our attention onthe “standard” treatment of the spin-orbit interaction in Wien2k.

The scalar relativistic bands are first calculated by setting up and diagonalizing the secular equationderived from (3.42) and (3.43). As a result, one obtains the eigenvalues εσkn and their correspondingeigenfunctions ϕσkn(r):

Hϕσkn(r) = εσn(k) ϕσkn(r) (3.46)

where k is the momentum, n the band index and σ the spin degree of freedom.6 The spin-orbitcoupling is then taken into account by using the method described hereafter: a second variationalsecular equation is set up with the scalar relativistic orbitals (for both spins) – ϕ↑

kn(r) and ϕ↓kn(r) –

as basis functions. This leads to a new secular equation where the Hamiltonian HSO and the overlapmatrices are given by:

〈ϕσkn|ϕτkm〉 = δnm δστ (3.47)

〈ϕσkn|H0|ϕτkm〉 = εσn(k) δnm δστ (3.48)

〈ϕσkn|HSO|ϕτkm〉 = HSOnσ,mτ (k). (3.49)

One thus obtains the following equation:

∀n, σ∑

m,τ

[εσn(k) δnmδστ +HSO

nσ,mτ (k)]zνmτ = εkνz

νnσ (3.50)

where ν runs over both spin and orbital indices, ν = ν(n, σ) with ν bijective. By solving this matrixequation, one finally gets the eigenvalues ενk and the corresponding eigenvectors:

ψkν(r) =∑

n,σ

zνnσϕσkn(r). (3.51)

Although the spin-orbit term couples the spin-up and spin-down wavefunctions, the wavefunctionsψkν(r) are still of the Bloch form:

ψkν(r) =∑

n

zνn↑ϕ↑kn(r) +

∑

n

zνn↓ϕ↓kn(r) = ψ+

kν(r) + ψ−kν(r)

=[u+kν(r) + u−kν(r)

]eik·r. (3.52)

As a result, the charge density can also be decomposed as follows:

ρ(k) =∑

ν occ.

〈ψkν |ψkν〉 =∑

ν occ.

[〈ψ+

kν |ψ+kν〉+ 〈ψ−

kν |ψ−kν〉]= ρ+(k) + ρ−(k) (3.53)

and a calculation with spin-orbit coupling can thus be performed almost “transparently” as a usual cal-culation with spin densities, by just adding a routine which includes this second variational treatment.The cycle in Wien2k, in which the spin-orbit interaction may be included self-consistently, is describedin Appendix C.

6As we have already said, the spin is a good quantum number within the scalar relativistic approximation.

3.3. IMPLEMENTATION OF THE SPIN-ORBIT COUPLING (SO) IN LDA+DMFT 53

3.3.2 Consequences on the definition of the Wannier projectors and on DMFTequations

According to equation (3.52), it is convenient to choose the basis |ψ+νk〉, |ψ−

νk〉 to describe the latticequantities in the LDA+DMFT implementation, since this is the natural output of the Wien2k code. Anew set of index i = +,− must thus be introduced to extend the formalism presented in section 2.3.The Wannier projectors can then be written as follows:

[Pα,σlm,ν

]+(k) = 〈χα,σlm |ψ+

kν〉 and[Pα,σlm,ν

]−(k) = 〈χα,σlm |ψ−

kν〉 (3.54)

where we remind that ν runs over both spin and orbital indices. In addition, the Green’s function andthe self-energy of the solid are now related by:

[G(k, iωn)]

−1ijνν′

= (iωn + µ− εkν)δνν′δij − Σijνν′(k, iωn) with i, j = +,−. (3.55)

In the following, we will consider the Wannier projectors as “spinors” in the “initial” Bloch basis|ψνk〉:

Pα,σlm,ν(k) =

[Pα,σlm,ν

]+(k)

[Pα,σlm,ν

]−(k)

(3.56)

whereas the Green’s function and the self-energy elements will be written as spin-matrices:

Gνν′(k, iωn) =

(G++νν′ (k, iωn) G+−

νν′ (k, iωn)

G−+νν′ (k, iωn) G−−

νν′ (k, iωn)

)

and

Σνν′(k, iωn) =

(Σ++νν′ (k, iωn) Σ+−

νν′ (k, iωn)

Σ−+νν′ (k, iωn) Σ−−

νν′ (k, iωn)

).

(3.57)

With this convention of notations, the equation (2.4) which defines the intial dynamical mean-fieldbecomes:

[G0(iωn)]σσ′

mm′ =∑

k,νν′

[Pα,σlm,ν(k)

]T[GKS(k, iωn)]νν′

[Pα,σ′

lm′,ν′(k)]∗. (3.58)

Similarly, the formula (2.8) which gives the expression of the local Green’s function merely can bewritten as:

[Gαloc(iωn)]

σσ′

mm′ =∑

k,νν′

[Pα,σlm,ν(k)

]TGνν′(k, iωn)

[Pα,σ′

lm′,ν′(k)]∗

(3.59)

and the expression (2.6) for the lattice self-energy correction is:

Σνν′(k, iωn) =∑

α,mm′

[Pα,σlm,ν(k)

]∗ [∆Σαimp(iωn)

]σσ′

mm′

[Pα,σ′

lm′,ν′(k)]T. (3.60)

Consequently, the equations of the DMFT loop are formally the same as in the case without spin-orbitcoupling (cf. expressions (2.45), (2.46) and (2.47)). However, the computations now involve matriceswhich are double in size.

The construction of the Wannier projectors is still done in two steps. The temporary Wannierprojectors are first calculated – separately for the |ψ+

νk〉 and the |ψ−νk〉 basis functions – with the

following expression:

[Pα,σlm,ν

]i(k) = [Aναlm(k, σ)]

i +LOMAX∑

nLO=1

cν,σLO

[Cα,LOlm Oα,σ

lm,l′m′

]iwith i = +,− (3.61)


the coefficients [Aναlm(k, σ)]i and

[Cα,LOlm Oα,σ

lm,l′m′

]ibeing direct outputs of the Wien2k program. They

indeed are the analog of the coefficients Aναlm(k, σ) and Cα,LOlm Oα,σlm,l′m′ given in (2.35) for the |ψσkν〉.

The orthogonalization is then performed in order to get the “true” Wannier projectors but this stepcannot be done independently for the two parts of the Wannier projector, since the overlap matrixOα,α

′

m,m′(k, σ) is defined by:

〈χα,σlm |χα′,σ′

lm′ 〉 =∑

k∈1BZ

[Oα,α

′

(k)]σ,σ′

m,m′(3.62)

with[Oα,α

′

(k)]σ,σ′

m,m′=

νmax(k)∑

ν=νmin(k)

〈χα,σlm |ψkν〉〈ψkν |χα′,σ′

lm′ 〉

=

νmax(k)∑

ν=νmin(k)

∑

i,j=+,−

[Pα,σlm,ν

]i(k)[Pα

′,σ′

lm′,ν

]j∗(k). (3.63)

By requiring it in the input file, it is still possible to get the projectors of the correlated orbitals incubic symmetry or in any desired basis. This option was also extended in order to define the Wannierprojectors related to states which mix spin up and spin down complex spherical harmonics, such asthe jeff = 1/2 and jeff = 3/2 states introduced at the end of section 3.1.

More details about the structure of the new implementation can be found in Appendix D and[110]. In addition, the interested reader can find in Appendix E how the Brillouin zone integration ofequation (3.58) and (3.59) are performed7.

3.4 Summary: Our “LDA+SO+DMFT” implementation within theLAPW framework

Our LDA+SO+DMFT implementation within the LAPW framework is summarized in figure 3.2. Itextends the previous LDA+DMFT implementation of Aichhorn et al. [1] so that it may take intoaccount the spin-orbit coupling in the construction of the Wannier orbitals which define the localimpurity problem. We present here only the “one-shot” approach of the cycle, some developments arecurrently in progress to implement the complete LDA+SO+DMFT cycle.

i) The DFT calculation with Wien2k

The electronic structure calculations are performed using the Wien2k package, an all-electron full-potential LAPW method (cf. section 2.2). The spin-orbit interaction is introduced in the Kohn-Shamequations as explained in part 3.3.1.

ii) Calculating the Wannier projectors with dmftproj

The correlated orbitals are built from the (L)APW+lo basis in the interfacing program called dmftproj.This program calculates the Wannier projectors P

α,σlm,ν(k) by following the procedure introduced by

Anisimov et al. [10], which is extended to take into account the spin-orbit interaction (cf. part 3.3.2).

7It is indeed possible to get an expression similar to (2.57) but one must then apply the symmetry operations of theShubnikov magnetic space group of the compound on the unsymmetrized Green’s function calculated in the irreducibleBrillouin zone.

3.4. SUMMARY: OUR “LDA+SO+DMFT” IMPLEMENTATION 55

Figure 3.2: Our “one-shot” implementation of LDA+SO+DMFT withinn the LAPW framework:

The DFT-LDA+SO calculation is performed within Wien2k [23]. The spin-orbit interaction isincluded in the Kohn-Sham equations with a second variational treatment. One then gets the eigenval-ues εkν and the Bloch states |ψ+

kν〉 and |ψ−kν〉 (i).

The correlated orbitals are then defined and the corresponding Wannier projectors Pα,σlm,ν(k) are

constructed in the interfacing program dmftproj (ii) in order to perform the DMFT loop. The latter(iii) consists in:- solving the effective impurity problem for the impurity Green’s function Gimp with a CTQMC solver[165], hence obtaining an impurity self-energy Σimp;- combining the self-energy correction with the Green’s function of the solid G(iωn) in order tocalculate the local Green’s function Gα

loc(iωn) – cf. equations (3.55), (3.59) and (3.60) –;- finally obtaining an updated dynamical mean-field G0 for the impurity problem.Once the DMFT loop has converged, the chemical potential is updated and the spectral density A(k, ω)can be calculated (iv) by using the projectors Θα

lmνj(k), which were also built while running dmftproj.


iii) The DMFT self-consistent loop

The DMFT self-consistent loop relies on the same formalism as previously introduced in section 2.1.However, taking into account the spin-orbit coupling modifies slightly the equations (cf. expressions(3.55), (3.59) and (3.60)).

To solve the impurity problem, we use the strong-coupling version of the continuous-time quantumMonte Carlo (CTQMC) method [165]. This method is based on a hybridization expansion and hasproved to be a very efficient solver for quantum impurity models in the weak and strong correlationregime. Moreover, this solver allows us to address room temperature (300 K) without any problems.

Sources of errors

The approximations performed by using the LDA+DMFT theory were previously presented in sec-tion 2.1. However, various other sources of errors were introduced with our choice of impurity solver.These numerical errors, which are rather hard to control, are the following:

Monte Carlo statistical errors: These are the statistical errors of the impurity solver itself. Theyare very well under control and can be checked by error estimations and extension of the runtimeof the code. These errors are qualitatively very different from the errors of other impurity solverscommonly used. Whereas the iterated perturbation theory (IPT) – or Hubbard-I – solvers useadditional approximations for the solution of the impurity problem, the only source of errors inour case is statistical Monte Carlo errors.

Numerical discretization errors: These stem from the fact that even within the continuous-timesolvers, the dynamical mean field G0(τ) has to be discretized. The magnitude of these errors ishowever negligible, and Fourier transform errors are well under control with the techniques used[61].

Analytical continuation: As explained in part 1.4.3, CTQMC simulations are restricted to the imag-inary time domain. In order to obtain quantities which are directly accessible to the experiment,like the spectral function A(k, ω), the imaginary time quantities have to be analytically contin-ued to the real axis. This process involves the inversion of an ill-conditioned matrix. Maximumentropy methods determine the most probable spectral density by using the concept of Bayesianinterference [78]. This induces errors which are hard to control and require careful analysis ofcovariance information.

iv) Post-processing: Calculating the spectral function A(k, ω)

As just mentioned above, an analytic continuation is needed in order to obtain results such as thespectral function A(k, ω). In our implementatioin, we choose to perform the analytic continuation ofthe impurity self-energy using a stochastic version of the maximum entropy method [19]. Moreover, anew set of projectors has to be built.

Introduction of the Θ-projectors

In order to calculate quantities for a given atom α and a particular orbital (spin) character lm (σ) –such as the spectral functions Aσαlm(k, ω) – , a set of projectors called “Θ-projectors” was built. Con-trary to the previously introduced Wannier projectors Pα,σlm,ν(k), their definition is not restricted to thecorrelated orbitals only. The formalism of these Θ-projectors was initially introduced by Aichhorn etal.. [1] We have also extended it so that they may take into account spin-orbit corrections.

3.4. SUMMARY: OUR “LDA+SO+DMFT” IMPLEMENTATION 57

As we have explained in section 2.2, inside the muffin-tin sphere SαMT associated to the atom α wecan write ψσkν(r) as:

ψσkν(r) =

lmax∑

l=0

+l∑

m=−l

[Aναlm(k, σ)u

α,σl (rα, Eα1l) +Bνα

lm(k, σ)uα,σl (rα, Eα1l) + Cναlm (k, σ)uα,σl (rα, Eα2l)]Y lm(r

α)

=

lmax∑

l=0

+l∑

m=−l

[Aναlm(k, σ) u

α,σlm,1(r

α) +Bναlm(k, σ) uα,σlm,2(r

α) + Cναlm (k, σ) uα,σlm,3(rα)]

(3.64)

with the notation:

uα,σlm,1(rα) = uα,σl (rα, Eα1l)Y

lm(r

α) , uα,σlm,2(rα) = uα,σl (rα, Eα1l)Y

lm(r

α)

and uα,σlm,3(rα) = uα,σl (rα, Eα2l)Y

lm(r

α).(3.65)

The orbital character α,l,m,σ thus contributes in the eigenstates ψσkν(r) through three terms.

However, the basis uα,σlm,ii=1,2,3 is not orthonormalized as already mentioned in 2.3. To makethe calculations easier, we introduce an orthonormal basis set φα,σlm,jj=1,2,3 for each atomic orbital(l,m). These orbitals are defined from the initial basis uα,σlmii=1,2,3 as follows:8

∀i uα,σlm,i(rα) =

3∑

j=1

cijφα,σlm,j with C =

1 0 〈uα,σlm,1|uα,σlm,2〉

0 〈uα,σlm,2|uα,σlm,2〉〈uα,σlm,2|u

α,σlm,3〉

〈uα,σlm,3|uα,σlm,1〉〈uα,σlm,3|u

α,σlm,2〉 1

12

. (3.66)

We can then rewrite (3.64) as:

ψσkν(r) =

lmax∑

l=0

+l∑

m=−l

3∑

j=1

Θα,σlmνj(k)φ

α,σlm,j(r

α). (3.67)

The matrix elements Θα,σlmνj(k) are the “Θ-projectors”, which are thus defined by:

Θα,σlmνj(k) = 〈φα,σlm,j |ψσkν〉 = Aναlm(k, σ)c1j +Bνα

lm(k, σ)c2j + Cναlm (k, σ)c3j . (3.68)

When the spin-orbit interaction is introduced, the definition of the projectors Θα,σlm,νj(k) must be

extended to “spinor” projectors Θα,σlmνj(k), in a similar fashion as previously described for the Wannier

projectors

Θα,σlmνj(k) =

(〈φα,σlm,j |ψ+

kν〉〈φα,σlm,j |ψ−

kν〉

). (3.69)

For the sake of simplicity, the Θ-projectors were introduced here in the complex spherical harmonicsbasis. As for the Wannier projectors, it is of course possible to get the Θ-projectors in any desiredbasis – even one which mixes spin up and spin down complex spherical harmonics, for a calculationincluding the spin-orbit coupling.

8We remind that the functions uα,σl (rα, E)Y l

m(rα) are normalized to 1 for each value of E by definition. As a result,〈uα,σ

lm,1|uα,σlm,1〉 = 〈uα,σ

lm,3|uα,σlm,3〉 = 1 and 〈uα,σ

lm,1|uα,σlm,2〉=0.


Calculation of A(k, ω)

By definition, the spectral function A(k, ω) is given by:

A(k, ω) = − 1

πIm [G(k, ω)] . (3.70)

As a result, the spectral function of a given atom α with orbital character (l,m) and spin σ, is finallyobtained through the following formulas:

• in a calculation without spin-orbit interaction

Aα,σlm (k, ω) = − 1

πIm

∑

νν′

3∑

j=1

Θα,σlmνj(k) G

σνν′(k, ω + i0+)

[Θα,σlmν′j(k)

]∗ (3.71)

• in a calculation which includes the spin-orbit corrections

Aα,σlm (k, ω) = − 1

πIm

∑

νν′

3∑

j=1

[Θα,σlmνj(k)

]TGνν′(k, ω + i0+)

[Θα,σlmν′j(k)

]∗ . (3.72)

We have now developed a new set of tools which can be used within the LDA+DMFT schemeto study compounds with significant spin-orbit terms. The first application performed was on theparamagnetic insulating phase of strontium iridate (Sr2IrO4).

Part II

The paramagnetic insulating phase ofStrontium Iridate Sr2IrO4

59

Chapter 4

Short review on strontium iridate Sr2IrO4

Although strontium iridate (Sr2IrO4) was first synthesized in 1956 by Randall et al. [136], it has takenalmost 40 years for the scientific community to draw some attention to this compound. More precisely,this dates back to 1994 when superconductivity was discovered in strontium ruthenate (Sr2RuO4)[109]. Since both the ruthenate and iridate compounds have similar crystallographic structure, therewas hope to shed light onto this unconventional state by studying also the iridate counterpart. An-other reason for renewed interest was to better understand the magnetic properties of the isostructuralcuprates, such as La2−xBaxCuO4, discovered in the late eighties [20].

Experimental studies based on different techniques have all come to two main conclusions:

• Sr2IrO4 is insulating at all temperatures, although it contains an odd number of electrons performula unit.

• It exhibits a canted-antiferromagnetic order below 240 K with a ferromagnetic momentof 0.023 µB/Ir, which is very small in comparison to the expected value for such a compound (ofabout 1 µB/Ir).

This was to some extent puzzling, because the bandwidth of Sr2IrO4 is fairly large, and the Coulombinteraction for a 5d element is expected to be small, meaning that the Mott localization of electronsshould be not very effective.

In 2008 Kim et al. [84] proposed a solution to this problem by emphasizing the role of the spin-orbit coupling in this compound : the insulating state of this material would indeed result from thecooperative interaction between the spin-orbit coupling and electronic correlations. This picture, called“spin-orbit driven Mott insulator ”, was quickly confirmed by resonant X-ray scattering [85] and hasalso given a framework to understand the canted-antiferromagnetic phase of Sr2IrO4 [79]. The purposeof this chapter is to review the existing experimental and theoretical works on Sr2IrO4, in order to putour calculations, which are presented in chapter 5, into context.

4.1 Crystal structure of Sr2IrO4

In 1956, Sr2IrO4was synthesized through a solid-state reaction “between iridium metal powder andstrontium oxide, carbonate, nitrate or hydroxide at 1200 C ” [136]. With this technique and its vari-ations [34, 72], polycrystalline samples (or pellets) which are “hard and black ” [135] are produced.Since 1998, another growth technique has also been developed with a flux method using strontiumchloride SrCl2 flux [31, 85]. With this new way of synthesis, one gets plate-like single crystals withtypical dimension of 1× 1× (0.1− 0.5) mm3. The magnetic and transport properties of single crystalsand polycrystalline samples are not entirely the same. In the following, we will explicitly specify the

61

62 CHAPTER 4. SHORT REVIEW ON STRONTIUM IRIDATE

nature of the samples used for the experiments, if a difference in their behavior was noticed.

The crystal structure of Sr2IrO4 was first described to be of K2NiF4-type, similar to Sr2RuO4 or thehigh-temperature superconductor La2−xBaxCuO4 [136]. This means that the compound is a layeredperovskite with planes of Ir-O2 and Sr-O. Moreover, it implies that iridium ions are surrounded by sixoxygen atoms, forming then IrO6 octahedra which are aligned along the crystal axes a and b.

However, electron diffraction measurements with Rietveld refinement [34] were carried out on theSr2Ru1−xIrxO4 system in 1994 and revealed weak super-lattice reflections, indicating that Sr2IrO4

has a lower symmetry. These results were corroborated by some further experiments (with powderX-ray diffraction or neutron diffraction) [41, 72, 88, 135, 156], which confirmed that crystallographicdistortions take place in this material: the corner-shared IrO6 octahedra are not well-aligned butare alternately rotated clockwise and anticlockwise around the c-axis of the crystal by about 11.This tilting of the octahedra lowers the symmetry space group of Sr2IrO4 from I4/mmm to I41/acd.Furthermore, DFT calculations performed by Cosio Castaneda et al. [39] have also confirmed that theI41/acd symmetry is more stable than the I4/mmm symmetry.

Figure 4.1: Conventional unit cell of Sr2IrO4. The green spheres stand for the strontium ions (Sr), thegolden ones for iridium (Ir) and the red ones for oxygen (O). The corner-sharing IrO6 octahedra arealternately rotated clockwise and anticlockwise around the c-axis by about 11. From [88]

The conventional unit cell of Sr2IrO4 is depicted in figure 4.1. The structural parameters at 10 Kand at room temperature can be found in [41, 72, 145]. The rotation of the IrO6 octahedra decreaseswith temperature, from 11.72 at 10 K to 11.36 at room temperature according to Huang et al. [72].The a and c axes have been reported many times with different values, ranging from 5.4921 to 5.4994 Åand from 25.766 to 25.798 Å respectively. This discrepancy in the lattice parameters has been ex-plained by the oxygen non-stoichiometry in the different samples, since the control of this stoichiometryduring the synthesis is difficult [88].

In figure 4.1, the IrO6 octahedra are all rotated with respect to each other, consistently with the con-straints imposed by the I41/acd symmetry. However, each layer containing the IrO6 octahedra is wellseparated from another by two Sr-O planes, which allow to consider Sr2IrO4 as a quasi-bidimensionalcompound. As a result, Huang et al. [72] proposed that the Ir-O2 layers may be uncorrelated and therotation of IrO6 octahedra in a layer can be independent of the – clockwise or anticlockwise – config-uration observed in its adjacent layers. Using neutron powder diffraction, they indeed confirmed that

4.2. EXPERIMENTAL EVIDENCE FOR AN INSULATING STATE 63

disorder occurs quite significantly in the compound: about 83% of the oxygen ion position of a layerare generated from those of the adjacent layers by the operation of the 41 axis and about 17% of themare uncorrelated to the oxygen atoms of the adjacent layers by any symmetry operation of the spacegroup. However, these results were obtained on polycrystalline sample, and no similar observationswere reported in studies on single crystals.

4.2 Experimental evidence for an insulating state

As mentioned in the introduction it was surprising to find Sr2IrO4 to be an insulator, since thiscompound has an open 5d shell. Nevertheless, many different probes give evidence for its insulatingbehavior.

4.2.1 Transport measurements

As usual, the electric transport properties are highly sample-dependent, which is due to the differentsynthesis routes resulting in different sample qualities. However, all the experiments agree that Sr2IrO4

exhibits an insulating behavior and a significant anisotropy between the a/b and the c directions.Moreover, no anomaly at the magnetic transition, which occurs at TM = 240 K, was reported. Toillustrate these main features, figure 4.2 displays the temperature dependence of the electrical resistivityfor the single crystal synthesized by Kim et al. [85].

Figure 4.2: Temperature dependence of in-plane resistivity (ρab) and out-of-plane resistivity (ρc).From [85].

At room temperature, the resistivity is estimated between 4 Ω.cm [87] and 10 Ω.cm [88]. The tem-perature dependence of the resistivity can not be fitted easily to a simple model. However, most authorsassume an Arrhenius-type behavior ρ(T ) = ρ0 exp(Ea/kBT ), which is characteristic for semiconductor-like behavior. Since this model is not well-satisfied in the whole temperature range, some fittings tothis model are reported by restriction to a certain range of temperatures:

• Shimura et al. [145] found Ea = 0.06 eV by considering only the temperature region below 200 K,

• Kini et al. [87] fitted the resistivity with ρ0 = 1.553 Ω.cm and Ea = 430.7kB = 0.037 eV in themiddle temperature range [110 K;190 K],

• Kim et al. [85] reported an activation energy gap Ea of 0.070 eV,

• Fisher et al. [52] found a regime of constant activation energy with Ea = 0.056 eV between 54and 205 K,


• Cosio Castaneda et al. [39] estimated the activation energy to be Ea = 0.047 eV in the tempera-ture region [220 K;300 K].

All the results give the same order of magnitude but it is commonly accepted that the true gap in thiscompound is much larger than the soft gap estimated by these electrical resistivity measurements.

For the sake of completeness, we mention that a very detailed study of the temperature dependenceof the electrical resistivity can be found in [87]. Particularly, for a low temperature range, a behaviorof the type ρ(T ) = A exp(T0/T )

ν with ν = 1/4 is found for the compound. This can be associated toa three-dimensional various-range hopping of carriers between states localized by disorder. A similarbehavior at low temperatures is mentioned in [31, 39].

4.2.2 Optical conductivity

The optical conductivity as reported in [84, 115, 116, 117] gives a more direct estimation of the gapthan the analysis presented previously. For example, Moon et al.[117] measured the optical reflectivityR(ω) at 100 K and at room temperature between 5 meV and 30 eV, the conductivity σ(ω) beingobtained by Kramers-Kronig transformation. Figure 4.3 displays the ab-plane optical conductivity ofSr2IrO4. Above 1.5 eV, the p-d charge transfer transitions observed are those from O 2p to dxy (dxz,dyz) for peak A, to d3z2−r2 for peak B and to dx2−y2 for peak C.

Figure 4.3: In-plane optical conductivity spec-tra σ(ω) of Sr2IrO4 at room temperature. Threepeaks labeled A,B and C can be observed above2.0 eV. From [117]

Figure 4.4: Zoom on the double peak structure,α and β, below 2 eV in the optical conductivityσ(ω) of Sr2IrO4at 100 K. From [84]

An optical gap of about 0.3 eV can be noticed at room temperature, the sharp spikes below 0.1 eVare due to optical phonon modes. The double peak structure, marked as α and β in figure 4.4 can notbe explained by a simple model invoking the 5d orbitals of iridium. As we will show later, spin-orbitcoupling is crucial for the explanation.

4.2.3 Spectroscopy measurements

With Angle Resolved Photo-Emission Spectroscopy (ARPES), one can investigates almost directly theelectronic structure of a compound. Figure 4.5 presents the results obtained by performing ARPESon single crystals of Sr2IrO4. These spectra were obtained by Kim et al. [84] at 100 K from sampleswhich were cleaved in situ under vacuum of 1 × 10−11 Torr. The energy distribution curves in panel(a) display the band features, whereas intensity maps at binding energies EB = 0.2, 0.3, and 0.4 eVare shown panels (b)-(d).

4.2. EXPERIMENTAL EVIDENCE FOR AN INSULATING STATE 65

Figure 4.5: ARPES spectra along Γ −M − X − Γ (panel a) and ARPES intensity maps at bindingenergies EB = 0.2, 0.3, and 0.4 eV (panels b,c and d). From [84]

It is obvious that there is no band crossing the Fermi level (which would be an indication for ametallic state). The gap is roughly consistent with the optical gap estimation presented previously.For the sake of completeness, we also mention that ultraviolet photoelectron spectroscopy and X-rayphotoelectron spectroscopy were also carried out on the material in [135]: the measurements confirmedthat Sr2IrO4 is an insulator with a gaped density of states (DOS).

4.2.4 Heat properties

Figure 4.6: Temperature dependence of the spe-cific heat (Cp) of Sr2IrO4. From [87]

Figure 4.7: Temperature dependence of the See-beck coefficient (S(T )) of Sr2IrO4. From [87]

Information about an energy gap can also be extracted from specific heat measurements. In fig-ure 4.6 the specific heat Cp(T ) of Sr2IrO4 is displayed as a function of the temperature. The specificheat is completely dominated by Debye-phonon, that is to say by lattice excitations, and can bemodeled by Cp(T ) = γ0.T + βT 3 with γ0, the Sommerfeld contribution, being of the order of 1.8mJ.mol−1.K−2 [33, 87]. Such a small value is in good agreement with the presence of a gap in Sr2IrO4.


1

A complete study about the Debye temperature can be found in [87]. Whereas it can not be observedon figure 4.6, recently Chikara et al. [38] measured a tiny specific heat anomaly ≈ 4 mJ.mol−1.K−2

at TM = 240 K, which was interpreted as a very small entropy change at the magnetic transition.The thermal conductivity κ(T ) was also studied by Kini et al.[87]. Their results confirm that heatconductivity in Sr2IrO4 is mainly due to lattice excitations.

The variation of the thermoelectric power – or Seebeck coefficient S(T ) – of Sr2IrO4 as functionof temperature is displayed in figure 4.7 [38, 88, 52]. The value of S(T ) is positive, which is due toa hole conduction. A broad maximum, whose value ranges from 270 to 320µV.K−1 depending on thesample, is noticed around 100-150 K, before a decrease to a value of about 120 µV.K−1 at 300 K. Noanomaly was ever reported at the magnetic transition temperature. Some attempts to estimate thegap of Sr2IrO4 from this curve was made but results do not agree with each other: Klein et al. [88]find an energy gap of 100 K – ≈ 0.01 eV – whereas Fisher et al.[52] estimate the gap to be of the orderof 0.3 eV from a high-temperature range fitting.

4.3 Theoretical models for the insulating state

In 2000, Rama Rao et al. performed the first electronic band structure calculations for Sr2IrO4, byusing the tight binding linear muffin-tin orbitals (TB-LMTO) within the atomic sphere approximation(ASA) [135]. The results incorrectly predicted Sr2IrO4 to be a metal with a finite DOSat the Fermilevel. The failure of the LDA description indicates that electronic correlations are responsible for theinsulating behavior of this compound. Mott physics based on the Hubbard Hamiltonian was theninvoked to describe Sr2IrO4.

However, 4d and 5d-transition metal oxides are characterized by the larger spatial extent of theird-electron orbitals in comparison to their 3d counterparts – since the orthogonality to other orbitalscan not be entirely assured by the angular part of the wavefunction. This feature enhances the d-phybridization between the transition metal and the oxygen, which results in the formation of bandstructure with larger bandwidth in these compounds than in the 3d-transition metal oxides. The LDAcalculations performed in [79] confirm that the band structure of Sr2IrO4 is almost identical to that ofSr2RhO4, with wide t2g bands ranging from −2.5 to 0.5 eV.

In addition, since the 4d and – even more – the 5d states are delocalized, the electron correlationsare commonly expected to play a smaller role in these compounds. As a result, to set the HubbardHamiltonian which will describe Sr2IrO4, the on-site Coulomb parameter U must be smaller thanthe typical values used for 3d-transition metal oxides– in which U typically ranges from 5 to 7 eV– and those used for 4d-transition metal oxides. Furthermore, a recent constrained-RPA calculationperformed on Sr2RuO4 has estimated the value of U in this compound of about 2.3 eV [122]. InSr2IrO4, the magnitude of the U parameter is thus expected to be of 2 eV maximum. Nevertheless, asit can be seen from the LDA+U calculations performed in [84], such a value for U cannot lead to aninsulating state for Sr2IrO4.

1The Sommerfeld contribution γ0 is rigorously defined only in the case of a metallic compound and is then proportionalto the electronic density at the Fermi level. On the contrary, the expression of the specific heat for an insulator is ratherCp(T ) ∼ T−1/2 exp(−Egap/2kBT ). However, it is still possible to fit this expression linearly at low temperature: in this

case, one gets γ0 ∼ ∆T−1∫∆T

0τ−3/2 exp(−Egap/2kBτ)dτ . The small value of γ0 obtained for Sr2IrO4 implies that this

last expression holds for this compound.

4.4. MAGNETIC PROPERTIES 67

4.3.1 The “spin-orbit driven Mott insulating” model

The breakthrough in the understanding of the properties of Sr2IrO4 was done in 2008 by Kim et al.[84], where the picture of a “spin-orbit driven Mott insulator ” in Sr2IrO4 was suggested. The spin-orbit coupling energy scale is indeed comparable to those of the other interactions in this 5d transitionmetal oxide: the spin-orbit coupling constant is estimated to be ζSO ≈ 0.4 eV for iridium according to[53, 43, 161].

As a result, by taking into account the spin-orbit coupling in the LDA calculations, the t2g bandssplit into jeff = 1/2 doublet and jeff = 3/2 quartet bands, as we have explained in the chapter 3.1.Moreover, the latter are completely filled and one electron remains in the narrow jeff = 1/2 bands. AU parameter of 2 eV is then enough to reach the Mott insulating state, as shown by the LDA+SO+Ucalculations performed in [84]. The cooperative interaction between electron correlation and the spin-orbit coupling thus explains the insulating state of Sr2IrO4. Furthermore, the electronic structures ofthe 5d Ruddlesden-Popper series Srn+1IrnO3n+1 can also be understood by this phenomenon [116].

An experimental confirmation of this theoretical model was given by resonant X-ray scattering in2009 [85]. This technique was indeed used to probe the relative phases of the electronic state, whichcorresponds to the spin-orbit driven Mott insulating state in Sr2IrO4. Results have shown that theground state is very close to the jeff = 1/2 limit, hence validating this theory.

In addition, the spin-orbit driven Mott insulator can also explain the double peak structure, markedas α and β in figure 4.4 as follows:

• Peak α is an optical transition from the lower Hubbard band to the upper Hubbard band of thejeff = 1/2 states.

• Peak β is the optical transition from the jeff = 3/2 bands to the upper Hubbard band of thejeff = 1/2 states.

X-ray absorption spectra has confirmed this picture [84]. An orbital ratio xy : yz : zx = 1 : 1 : 1 withinan estimation error < 10% for the unoccupied t2g state was found, in good agreement with the ioniclimit of the jeff = 1/2 state (cf. equations (3.27)).

4.4 Magnetic properties

4.4.1 Experiments

Figure 4.8 displays the magnetic susceptibility χ(T ) = M(T )/H of Sr2IrO4 at H = 0.5 T along thetwo principal crystallographic directions.2 A clear ferromagnetic transition is observed at TM = 240 K,as already mentioned in [34, 38, 88, 145]. The large anisotropy of the magnetic susceptibility indicatesthat the easy axis is aligned with the a axis.

Above TM , the susceptibility can be fitted with a modified Curie-Weiss law χ(T ) = χ0+C/(T − θCW ),where C = NAµ

2eff/3kB, with the following parameters [39, 87]:

• A Curie-Weiss temperature θCW between 236 and 251 K,

• A temperature independent susceptibility χ0 of 1.5-8.8×10−4 emu.mole−1,

• An effective magnetic moment µeff of about 0.3-0.5µB/Ir.

2The data have been corrected for core diamagnetism with the value 1.06×10−4emu.mole−1.


The discrepancy in the result is due to the fact that the ferromagnetism is very weak. Whereas θCWis comparable to the magnetic ordering temperature, the range of variations of µeff is significantlylower than the expected Hund’s rule value of 1.73 µB/Ir for S=1/2. The temperature independentsusceptibility χ0 is likely due to a Van Vleck contribution [33], because of the insulating behavior ofSr2IrO4.

Figure 4.8: Magnetic susceptibility χ(T ) =M(T )/H of Sr2IrO4 at H = 0.5 T along thetwo principal crystallographic directions. Inset:∆χ−1 in function of the temperature for T > TM ,with ∆χ = χ(T )− χ0. From [31]

Figure 4.9: Isothermal magnetization M ofSr2IrO4 in function of the magnetic field H atT = 5 K. Inset: Isothermal magnetization alongthe a-axis for −0.5 ≤ H ≤ 0.5 T. From [31]

To be sure of the nature of the magnetic transition, magnetic hysteresis measurements (magneti-zation versus magnetic field) were performed at 5 K [31, 39, 41]. The observed hysteresis behavior,which can be seen on figure 4.9 is characteristic of a ferromagnetic compound. The saturation magneticfield was estimated more than 6.5 T and the magnitude for the ferromagnetic moment µferro of about0.023 µB/Ir. This value is far too small to attribute it to full ferromagnetically aligned spin 1/2 ions(corresponding to 1 µB). Moreover, the saturation moment µS is estimated of about 0.13-0.18 µB/Irat low field (H > 0.5 T) [31, 38, 39] and of about 0.03-0.045 µB/Ir between 4 T and 5 T [41, 87].In order to explain these low values of (saturation or effective) magnetic moment, the hypothesis of acanted antiferromagnetism was suggested in a early work by Crawford et al. [41].

Figure 4.10: Magnetic configuration (panel a) and Dzyaloshinskii-Moriya vectors in Sr2IrO4 (panelb). The blue arrows in (a) represent the local iridium moments, consisting of both spin and orbitalcomponents, in a canted antiferromagnetic configuration. The Dzyaloshinskii-Moriya vectors in (b)are aligned along the c-axis. From [79]

4.5. STILL OPEN QUESTIONS ABOUT SR2IRO4 69

4.4.2 Model for the canted-antiferromagnetism

In 2009, resonant X-ray scattering performed by Kim et al. [85] have confirmed that the magneticstructure of Sr2IrO4 is canted antiferromagnetic. In addition, based on the spin-orbit driven Mottinsulator theory, a microscopic model was built to explain the magnetic state of Sr2IrO4 below TM =240 K [77, 79]. It is indeed possible to consider an effective Hamiltonian based on the jeff = 1/2single-band Hubbard model:

H =∑

〈ij〉

∑

m,m′

tijmm′d†imdjm′ + U

∑

i

ni,+ 12ni,− 1

2(4.1)

where d†im stands for the |φjeff= 12,m〉 state (3.27) at the site i with m = ±1/2 and nim = d†imdim.

tijmm′ and U are then the effective hopping and on-site Coulomb parameter. By taking into accountthe rotation of the IrO6 octahedra, a spin dependent hopping term is generated and the followingjeff = 1/2-spin Hamiltonian can be derived:

Hspin =∑

〈ij〉

[I0Ji · Jj + I1JziJzj +Dij · Ji × Jj ] (4.2)

The first term is a conventional Heisenberg form of superexchange, the second and third terms arepseudodipolar and Dzyaloshinskii-Moriya antisymmetric exchange interactions. This last term ex-plains the canted antiferromagnetism of Sr2IrO4 with the ab plane, as depicted on figure 4.10. More-over, LDA+SO+U calculations [79] have confirmed that the magnetic configuration of Sr2IrO4 is acanted antiferromagnetic order with an angle close to the rotation angle of IrO6 octahedra. ThisDzyaloshinskii-Moriya type of magnetic interaction is thus the result of both the strong spin-orbitcoupling and the distortions. Since 5d orbitals of iridium atoms are extended, exchange terms favor-ing parallel spin configurations are negligible. Moreover normal superexchange bonding contributionis more important than corresponding ferromagnetic exchange, hence favoring an antiferromagneticarrangement. Because of the rotation of the IrO6 octahedra, the center of inversion located on theoxygen is removed and a non collinear antiferromagnetic ordering is then allowed.

4.5 Still open questions about Sr2IrO4

Despite the success of the spin-orbit driven Mott insulator model, in particular to explain the character-istics of the phase below TM = 240 K, some properties of Sr2IrO4 still remains out of the understanding.

4.5.1 Specificities of the electrical transport in single crystal

On the one hand, a non-trivial conducting behavior was reported for single crystal samples [31]: thecurrent-voltage (I-V) characteristics exhibit a current controlled negative differential resistivity for botha and c directions, which occurs at very low voltage and decreases with increasing temperatures. I-Vmeasurements were also carried out on polycrystalline samples [52]. Large deviations from linearitywere found at high fields for the dc current density-electric field (J-E) characteristics, while the pulsedJ-E characteristics have exhibited only weak non-linearity. Such a behavior seems to be attributed toan electrothermal effect – Joule heating – but more experimental works are necessary to draw a clearconclusion on this subject.

On the other hand, an unconventional giant magnetoelectric effect was recently observed in a singlecrystal of Sr2IrO4 [38]. It is characterized by a strongly peaked permittivity near an observed magneticanomaly at about 100 K and a large magnetodielectric shift that occurs near a metamagnetic transition.Contrary to current models, this effect however depends on strong spin-orbit coupling rather than themagnitude and spatial dependence of magnetization. Further investigations on this phenomenon areexpected in the future.


4.5.2 Temperature dependence of the optical gap

Figure 4.11 depicts the temperature-dependent optical conductivity σ(ω) of Sr2IrO4 [115]. As thetemperature increases from 10 to 500 K, the optical gap is estimated to change from 0.41 to 0.08 eV.At room temperature (300 K), its value is about 0.26 eV. Moreover, the change in the optical gapis largest near the magnetic transition temperature. Looking more precisely to the evolution of thestructure of the optical conductivity, the sharp peak α becomes broader and shifts to lower energywith increasing temperature. In addition, while the heights of the peaks α and β decrease, the spectralweight of the peak α increases and that of the peak β decreases as the temperature increases.

Figure 4.11: Temperature dependence of optical conductivity spectra σ(ω) of Sr2IrO4. As temperatureincreases, the peak α and β become broader and the optical gap decreases. From [115]

This decrease of the spin-orbit driven Mott gap has not been explained yet. The rate of decrease inthe gap with temperature is about four to five times larger than those of the semiconductors. Therefore,it can not merely be associated to a thermal effect. Furthermore, the simple lattice distortions cannot explain the large change in the bandwidth of Sr2IrO4 with temperature variation, according toLDA+SO+U calculations [115].

Summary

Sr2IrO4 is a 5d-transition metal oxide with a K2NiF4-type structure in which the corner-shared IrO6

octahedra are not well-aligned but are alternately rotated clockwise and anticlockwise around the c-axis of the crystal. This compound is insulating and undergoes a magnetic transition at TM = 240 K.Below TM = 240 K, Sr2IrO4 indeed exhibits a unusual weak ferromagnetism with reduced iridiummagnetic moments, which is attributed to a canted antiferromagnetic order.

These two surprising features can be explained by a cooperative interaction between electroniccorrelations and spin-orbit coupling: Sr2IrO4 can indeed be understood as a spin-orbit driven Mottinsulator whose gap – of about 0.26 eV at 300 K – lies between the lower Hubbard band and the upperHubbard band associated to its jeff = 1/2 states. Below TM = 240 K, the combined effect of the strongspin-orbit coupling and the distortions generates a Dzyaloshinskii-Moriya type of magnetic interaction,which results in the observed canted antiferromagnetism.

However, the spin-orbit driven Mott insulating model was only confirmed in the magnetic phase(T < 240 K) and is unable to explain the decrease of the optical gap with raising temperature. Beforetrying to explain this phenomenon, a band-structure calculation which includes both the spin-orbitcoupling and the electronic correlations must be carried out in the paramagnetic phase of Sr2IrO4. Ourwork, which is described in the following chapter, presents the results obtained by studying Sr2IrO4

within LDA+DMFT. They highlight that the joint effort of the spin-orbit coupling and the structuraldistortions is essential to trigger the Mott transition in Sr2IrO4.

Chapter 5

The Sr2IrO4 Mott insulator:Role of the spin-orbit coupling and thelattice distortions

As described in the previous chapter, at room temperature Sr2IrO4 is a paramagnetic insulator withan optical gap of about 0.26 eV [115] However, DFT calculations give a metal as displayed in figure 5.1.Indeed, strontium and oxygen ions are respectively in the state Sr2+ and O2−, which yields 5 electronson the Ir 5d orbitals by formula unit, an odd number which is incompatible with an insulating statein the band picture. The insulating state must thus be caused by electronic correlations, which mightseem surprising for a 5d element with strongly screened repulsion.

Z Γ M X Γ P N -3

-2

-1

0

1

2

Ene

rgy

(eV

) EF

Figure 5.1: Kohn-Sham band structure of Sr2IrO4 in its realistic structure (I41/acd symmetry). Thiscalculation performed with Wien2k takes into account the spin-orbit coupling corrections. Four bands(in red) cross the Fermi level, which implies a metallic description of the compound.

71

72 CHAPTER 5. THE SR2IRO4 MOTT INSULATOR

In this chapter, we demonstrate that the following features collaborate to yield an insulator: (i)band narrowing due to the distortions in the material, (ii) spin-orbit coupling, and (iii) the Coulombinteraction with additional Hund’s rule coupling. To achieve this goal, we use the LDA+DMFT toolboxas a numerical experiment in which we will subsequently turn on/off each of these features separately.This will enable us to understand why, in the real material, where all of these features are present, amoderate repulsion parameter U = 1.3 eV – as we will show – is sufficient to cause the transition toan insulator.

The scheme of our numerical experiment is presented on figure 5.2. There are thus four combinationswe consider:

- an “idealized ” description of Sr2IrO4, without distortions without spin-orbit coupling

- two “intermediate” states: an undistorted case with spin-orbit coupling and a distorted one withoutspin-orbit coupling

- a description which includes both the distortions and the spin-orbit coupling, which actually corre-sponds to the “realistic” Sr2IrO4.

The chapter is composed of two main parts:

• In the first section, the results of our DFT calculations are given. The reorganization of theband structure induced by taking into account the spin-orbit coupling and the distortions areconsidered, explaining the nature of the four narrow bands lying close to the Fermi level in theband structure of Sr2IrO4.

• In the second section, the results of our LDA+DMFT calculations are shown for each of the fourcases, for several values for the Coulomb parameter U. The critical value at which the metal-insulator transition occurs is determined and used to characterize the fragility of the metallicstate to correlations. In a nutshell, including the distortions reduces the bandwidth and inducesan orbital polarization, while the spin-orbit coupling lifts some of the degeneracies of the metal-insulator transition. All these effects are necessary to explain the insulating nature of Sr2IrO4.

5.1 Influence of the spin-orbit coupling and the distortions on elec-tronic correlations: Density functional study

5.1.1 Case 1: “Undistorted” Sr2IrO4 without spin-orbit coupling

We start by presenting the results of a DFT calculation performed on “undistorted ” Sr2IrO4 in which thespin-orbit coupling is neglected. The structure of the compound is then of K2NiF4-type, and its bandstructure is similar to that of strontium ruthenate (Sr2RuO4) or the high-temperature superconductorlanthanum-baryum cuprate (La2−xBaxCuO4. This will enable us to present some general electroniccharacteristics of Sr2IrO4, such as its low-energy properties dominated by its t2g-bands.

The unit cell and the chemical composition

According to X-ray and neutron diffraction studies [41, 72, 88, 135, 156], Sr2IrO4 has a tetragonalI41/acd space group. It corresponds to the symmetry of a K2NiF4-type compound in which thecorner-shared octahedra are not well-aligned along the crystallographic directions a and b, but arealternately rotated clockwise and anticlockwise around the c-axis. In the “undistorted ” structurethe tilting of the IrO6 octahedra are neglected and thus the body-centered tetragonal unit cell of a

5.1. EFFECTS OF SPIN-ORBIT & DISTORTIONS WITHIN DFT-LDA 73

Figure 5.2: Scheme of our investigation. By following the blue arrow, the distortions are tuned upto their realistic level (a rotation of about 11 around the c-axis); by following the green arrow, thespin-orbit corrections are introduced.

Figure 5.3: Conventional unit cell of Sr2IrO4 in the “undistorted ” K2NiF4-type structure (I4/mmmsymmetry). The blue spheres stand for the strontium ions (Sr), the golden ones for Iridium (Ir) andthe red ones for Oxygen (O). The covalent radius of each atomic species was used to set the size of therepresenting spheres. This picture was obtained with the software Xcrysden.


K2NiF4-type compound becomes suitable to describe of Sr2IrO4. The corresponding space-group isusually called I4/mmm and the corresponding primitive unit cell contains only one formula unit.

The conventional representation of the unit cell of Sr2IrO4 in this undistorted symmetry is depictedin figure 5.3. The compound is a stacking of Ir-O2 and Sr-O planes. Oxygen atoms which lie in thesame plane as iridium (strontium) atoms will be refer to as O1 (O2) in the following. We use the originallattice parameters from [136]:1 a = b = 3.89 Å and c = 12.92 Å. The crystallographic direction a,b,ccorresponds to the usual x,y,z axis. The atomic positions are given in the first column of table 5.1. Theionization states of each atomic species in the crystal are Sr2+, O2− and Ir5+. The iridium atoms thusaccommodate 5 valence electrons (by formula unit) in their 5d orbitals. More details on the electronicconfiguration of each species can be found in table 5.2.

Each iridium atom is surrounded by four in-plane O1 atoms and two apical O2 atoms, formingthen an octahedron (with O1IrO1 = O2IrO1 = 90). These corner-shared IrO6 octahedra are well-aligned along a and b direction ( IrO1Ir = 180). Because of this local cubic crystal field, the 5datomic orbitals of the iridium ions split into two eg and three t2g states. The octahedra are almost notelongated along the c axis: Ir-O1=1.945 Å and Ir-O2=1.951 Å. Consequently, the tetragonal crystalfield induced by this distortion can be neglected in a first approach and no other splitting is thusexpected in the 5d states2. The 5 valence electrons will thus occupy the three degenerate t2g orbitals(dxy, dxz and dyz) and the two degenerate eg orbitals (d3z2−r2 and dx2−y2), with higher local energy,will be empty.

Technicalities of the DFT calculation

We have performed a DFT calculation on this undistorted description of Sr2IrO4 using the Wien2k pack-age [23]. The main principles of this all electron full potential LAPW method have been presented insection 2.2. The calculations are done while using the LDA for the treatment of the exchange-correlationterm. Standard parameters of Wien2k code are used: more precisely RMT .Kmax = 7, lmax = 10 andlns,max = 4. The irreducible Brillouin zone is sampled with 99 k-points with a tetrahedral mesh.

The muffin-tin radii chosen for each atomic species are presented in the second column of table 5.1.The energy threshold, which defines the boundary between the well-localized (core) and the delocal-ized (semi-core and valence) electronic states, is set at −7.5 Ry. This implies the categorization ofthe electronic states of each atomic species as described in table 5.2. The total semi-core and valencestates considered during the calculation is then of 46+29=75 electrons by unit cell.

The band structure is plotted along the k-path depicted in figure 5.4. The conventional Brillouinzone is used. This k-path is chosen such that it contains the high-symmetry directions of the system.In the following, our attention will be focused on the [ZΓ] direction – to investigate the kz dispersionof the bands and then check the two-dimensional character of the compound – and the [ΓMXΓ] path– to understand the kx and ky dispersion properties.

Kohn-Sham band structure of undistorted Sr2IrO4

Our LDA calculation without spin-orbit coupling predicts a metallic nature for Sr2IrO4 in the undis-torted symmetry. As observed on figure 5.5, the total density of states (DOS) has indeed a finite value

1As mentioned in chapter 4, the distortions were indeed not detected by X-ray diffraction when Sr2IrO4 was synthesizedfor the first time in 1956. As a result, the compound was described in the undistorted symmetry, which makes the latticeparameters of this article particularly appropriate for our study.

2We will see that this approximation can not be used anymore in subsection 5.1.2. Indeed, despite its small value.this tetragonal crystal field must be taken into account to construct the local jeff = 1/2 and jeff = 3/2 basis out of theKohn-Sham band structure of undistorted Sr2IrO4 with spin-orbit.


Figure 5.4: Conventional Brillouin zone for Sr2IrO4 in the undistorted symmetry. The high symmetrypoints are Γ(0, 0, 0), Z(0, 0, 1), M(0.5, 0, 0), X(0.5, 0.5, 0), P (0.5, 0.5, 0.5) and N(0.5, 0, 0.5). The greenarrows depict the k-path along which bands are plotted in the following.

Atomic species Position Muffin-tin radiusin reduced coordinates in Bohr radius a0 = 0.529Å

Sr (0,0,0.347) 2.38 a0Ir (0,0,0) 1.94 a0O1 (0,0.5,0) 1.72 a0O2 (0,0,0.151) 1.72 a0

Table 5.1: Description of the structural parameters used for our DFT calculation performed on Sr2IrO4

in the undistorted I4/mmm symmetry.

Atomic species Core states Semi-core states Valence statesdescription electrons description electrons description electrons

Sr2+ [Ar]3d10 28 e− 4s24p6 8 e− 5s04d05p0 0 e−

Ir [Kr]4d10 46 e− 5s25p64f14 22 e− 6s05d56p 5 e−

O2−1 , O2−

2 [He]=1s2 2 e− 2s2 2 e− 2p6 6 e−

Total number of electrons 110 e− 46 e− 29 e−

Table 5.2: Electronic configuration of the atoms in Sr2IrO4 and repartition between the core, semi-coreand valence states as used during our DFT calculations.


at the Fermi level. To be more precise, in the Kohn-Sham band structure depicted on figure 5.6, fourbands cross the Fermi level.

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2Energy (eV)

0

2

4

6

8

10

12

14

DO

S (s

tate

s.eV

-1)

TotalIrSrO

1

O2

Figure 5.5: Total and partial desnity of states (DOS) of Sr2IrO4 in the undistorted symmetry withoutspin-orbit coupling. The Fermi level is materialized by the dotted vertical line.

As shown on figure 5.3, the structure of undistorted Sr2IrO4 is strongly anisotropic. Each layercontaining the IrO6 octahedra is indeed well separated from another by two Sr-O planes, correspondingto a distance of 2.56 Å. This implies a quasi-two dimensional character for the compound. By lookingalong the [Z Γ] path on figure 5.6, the kz dependence of the bands can be studied. Since almost nodispersion is observed in this direction, the main features of the Kohn-Sham band structure of undis-torted Sr2IrO4 can be understood by considering only a single two-dimensional plane of IrO6 octahedra.

On figures 5.5 and 5.6, only the valence bands – the bands which accommodate the 29 valenceelectrons – are displayed. They are exactly 17, as the number of atomic valence states in a formulaunit (the 5 Ir 5d levels and the 4×3 O 2p levels). To give a general description of the “nature” of thesebands, a simple tight-binding model can be developed: the IrO6 plane is restricted to a two-dimensionalsquare lattice with an effective atom in each node, whose “atomic levels” are the molecular orbitals ofan IrO6 octahedron. The pattern 5.7 summarizes the main structure of the molecular orbital diagramobtained for an IrO6 octahedron. By drawing a parallel between this picture, the Kohn-Sham bandstructure of figure 5.6 and the partial DOS associated to the iridium and oxygen atomic characters –displayed on figure 5.5 –, it is then possible to give the following interpretation:

• Between −9.5 eV and about −5 eV, the bands come from the bonding eg(σ) and t2g(π) molecularorbital obtained from the Ir 5d and the O 2p states. They have mostly an O 2p character andfor convenience, we will refer to them in the following as the “oxygen-bands”.

• Between −5 eV and −2.5 eV, the bands come from the non-bonding O 2p states, which explainswhy the partial DOS associated to the iridium character is almost zero in this range. We willalso refer to them in the following as “oxygen-bands”.

• Above −2.5 eV, the 5 bands (the 3 blue and the 2 red ones on figure 5.6) come from the anti-bonding eg(σ∗) and t2g(π∗) molecular orbital built from the Ir 5d and the O 2p states. They are


Z Γ M X Γ P N

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

Ene

rgy

(eV

)

EF

EF

Figure 5.6: Kohn-Sham band structure of Sr2IrO4 in the undistorted symmetry without spin-orbitcoupling. The lower boundary of the energy window is chosen such that the bands up to the Fermilevel contain the 29 valence electrons. In blue the t2g bands, in red the eg and in black the O2p bands.On ΓZ there is almost no dispersion, confirming the two-dimensional nature of the material.

Figure 5.7: The molecular orbitals in an IrO6 octahedron.


partially filled and have mainly an Ir 5d character. For convenience, we will refer to them as the“eg and t2g bands” in the following.

The atomic Ir 5d character from which each eg or t2g band derive has been plotted on figure 5.8.Each band has a pure character, which confirms that the cubic Ir 5d states are well-adapted to describethis band structure. The three bands which lie in the lower energy range – mostly below the Fermilevel – are associated to the Ir 5d t2g states and the two bands in the higher energy range – mainlyabove the Fermi level – to the Ir 5d eg states. Because of this one-to-one correspondence, we will nowcall each band with the name of its corresponding atomic state.

Z Γ M X Γ P N

-3

-2

-1

0

1

2

Ene

rgy

(eV

)

EF

EF

Figure 5.8: Kohn-Sham band structure of Sr2IrO4 in the undistorted symmetry without spin-orbitcoupling. The dxy band is represented in blue, the dxz and dyz in green, the dx2−y2 in red and thed3z2−r2 in yellow.

The shape of each t2g band along the k-path can be understood with a tight-binding approach basedon an effective model, in which iridium atomic orbitals are only involved. The cubic dxz orbitals on eachiridium site are perpendicular to the plane and form π bonds along the x-axis via the O1 2pz orbitals.The hopping terms along the y-axis can be neglected, the model is thus composed of independentunidimensional chains of iridium along x. Within this framework, the electron dispersion of the dxzband is of the following form:

εkxz = ε0xz − 2tπ cos(kx.a) (5.1)

where ε0xz is the effective local energy associated to the atomic-like dxz orbitals and tπ is the effectivehopping amplitude between them. A similar model can be developed for the dyz orbitals.

On the contrary, the dxy orbitals which lie in the plane require a two-dimensional pattern. Theyindeed form π bondings along the x-axis and the y-axis. However, the hopping between next-nearestneighbors must also be included so as to get the following electron dispersion of the dxy band:

εkxy = ε0xy − 2t[cos(kx.a) + cos(ky.a)

]− 4t′ cos(kx.a) cos(ky.a) (5.2)


-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2Energy (eV)

0

0.5

1

1.5

DO

S (s

tate

s.eV

-1)

Ir-d3z

2-r

2

Ir-dxy

Ir-dx

2-y

2

Ir-dxz

/dyz

Figure 5.9: Partial DOS associated to the Ir 5d eg and t2g characters for Sr2IrO4 in the “undistorted”symmetry without spin-orbit coupling. The Fermi level is materialized by the dotted vertical line.

where ε0xy is the effective local energy associated to the atomic-like dxy orbitals and t (t′) the hoppingamplitude between the nearest-neighbors (the next-nearest neighbors respectively). The larger band-width of the dxy band – almost twice wider than the one for the dxz and the dyz bands – thus comesfrom its two-dimensional character.

Similar effective models can also be developed to explain the shape of the two eg bands. This tight-binding approach is in good agreement with the structure of the partial DOS associated to each Ir 5dcharacter and represented in figure 5.9. The peak located at −0.1 eV in the partial DOS associated tothe dxy character can be identified as the Van Hove singularity of a two-dimensional band. Similarly,the two peaks, at −1 eV and at the Fermi level respectively, of the band dxz (or dyz) can be linked tothe two Van Hove singularities which can be found in a one-dimensional band.

Construction of a localized eg-t2g Wannier-type basis set for undistorted Sr2IrO4

In Sr2IrO4, the Ir-5d orbitals are considered to span the correlated subspace of the system. Sinceelectronic correlations play a more significant role on partially filled bands than on completely filled orempty ones, we construct the effective local impurity model used for the DMFT treatment from theWannier orbitals related to the five eg and t2g bands. To perform this, we use the scheme to buildWannier projectors which was described in the section 2.3. By choosing an energy window rangingfrom −3.5 eV to 0.6 eV, it is possible to associate a Wannier function to each band (in a similar pictureas in figure 5.8).

Their respective charges, which correspond to the filling of the band, are presented in the secondcolumn of table 5.3. The t2g bands are almost equally filled and accommodate 5 electrons per formulaunit. Furthermore, the eg bands of the system are almost empty, although the dx2−y2 band cross theFermi level.

Consequently, in the following we consider a “t2g bands” system with 5 electrons (by formula unit)in 3 t2g bands. Besides, in a calculation where electronic correlations will be better taken into account –such as a LDA+DMFT calculation –, the d3z2−r2 and dx2−y2 are expected to be shifted above the Fermi


Charge calculated from of the Wannier functionsthe partial DOS of the energy window [-3.5;0.6] eV

Ir 5d d3z2−r2 0.005 0.005Ir 5d dx2−y2 0.016 0.017Ir 5d dxy 0.731 1.612Ir 5d dxz/dyz 0.752 1.696

Table 5.3: Charge repartition between the eg and t2g atomic states (first column) and the eg andt2g bands (second column), between −3.5 eV and the Fermi level. The results of the first columnwere calculated by integrating the partial DOS calculated in Wien2k for the corresponding Ir 5d statefrom −3.5 eV to the Fermi level. The results of the second column were evaluated similarly but byconsidering the Wannier orbitals built in the energy window [−3.5; 0.6] eV. The Wannier orbitals areclearly “more than” just the Ir 5d orbitals.

level, which will remove completely the charge in them. Undistorted Sr2IrO4 has a large bandwidthof the dxy band (about 4.1 eV). As a result, an insulating state would emerge only if unphysicallylarge interaction was introduced. For realistic values of the Coulomb parameter U (below 2 eV for a5d-transition metal oxide), only moderately correlated metallic state, such as that of the Sr2RuO4, willemerge. More quantitative results are presented in section 5.2.

5.1.2 Case 2: Modifications in the Kohn-Sham band structure induced by thespin-orbit interaction

The second-variational method used in Wien2k to take into account the spin-orbit coupling was de-scribed in section 3.3. In our calculation, the spin-orbit corrections have been introduced only for theiridium atoms. Since the system is not spin-polarized, its symmetry is unchanged3 and the Kohn-Sham band structure can therefore be plotted along the same k-path (in the same Brillouin zone) aspreviously.

Kohn-Sham band structure of the undistorted Sr2IrO4 including the spin-orbit coupling

Our LDA calculation predicts also a metallic nature for undistorted Sr2IrO4 when the spin-orbit cou-pling is taken into account. To ease the comparison, the total DOS of the compound with and withoutspin-orbit coupling are both represented on figure 5.10. The major rearrangements of the structureoccur mostly between −2.5 eV and 1 eV, which corresponds to the energy range where eg and t2g bandspreviously lay. However, since Ir 5d states are also involved – but are in minority – in the formationof the oxygen bands which lay between −9.5 and −5 eV, some little changes can be noticed in thisenergy range too, but they will not be discussed in the following.

The Kohn-Sham band structure depicted on figure 5.11 reveals that still four bands cross the Fermilevel. Although the spin quantum number is not well-adapted to describe a system in which the spin-orbit coupling is taken into account, each band on figure 5.11 is still twice degenerate, because thespatial inversion is included in the undistorted symmetry (I4/mmm) and the system is paramagnetic4.As a result, each band still accommodates up to 2 electrons. The main changes have mostly affectedthe t2g bands, whereas the eg bands have essentially been shifted up by about 0.1 eV. By plotting theatomic Ir 5d character from which each eg and t2g bands derive, this first impression is confirmed:

3Taking into account the spin-orbit coupling does not modify the magnetic Shubnikov space group of the compound,as explained in Appendix E.

4This property was demonstrated in section 3.1.


-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3Energy (eV)

0

2

4

6

8

10

12

14

DO

S (s

tate

s.eV

-1)

with Spin-Orbit correctionswithout Spin-Orbit

Figure 5.10: Total DOS for Sr2IrO4 in the undistorted symmetry. The red curve includes the correctionsinduced by taking into account the spin-orbit coupling. The Fermi level is materialized by the dottedvertical line. The major change in the overall structure occurs close to the Fermi level between −2.5 eVand 1 eV.

Z Γ M X Γ P N

-3

-2

-1

0

1

2

Ene

rgy(

eV)

EF

Figure 5.11: Kohn-Sham band structure of Sr2IrO4 in the undistorted symmetry including the spin-orbit coupling. The eg bands (d3z2−r2 in yellow and dx2−y2 in red) are almost not affected by thecorrections and their atomic character is preserved. On the contrary, the t2g bands (in purple) aredeeply modified and mix from now on the dxy, dxz and dyz atomic characters along the k-path.


• Each eg band has indeed kept its own atomic Ir 5d character (d3z2−r2 and dx2−y2 – in yellow andred respectively on figure 5.11 – ). The spin-orbit coupling has thus not affected them and theobserved energy shift is actually the consequence of the new charge repartition between the t2gbands.

• The t2g bands are not in one-to-one correspondence with one Ir-5d t2g character anymore. Amixing of the three cubic states (dxy, dxz and dyz) can indeed be found in each band, whichimplies that the t2g cubic basis is not well-suited anymore to describe this new band structure.

Although figure 5.11 does not display the repartition of the dxy, dxz and dyz character along the t2gbands (in purple on the figure), the partial DOS associated to these atomic characters – and presentedin figure 5.12 – can confirm this last statement. First, the energy range of the dxz (or dyz) characterhas increased: for instance, its partial DOS has almost the same significant weight as that of dxy statebetween 0.2 eV and 0.75 eV, where only the “upper” t2g band lie. Furthermore, the Van Hove singu-larities associated to the t2g states have reduced significantly, which implies strong deviations with thetight-binding model based on the Ir 5d t2g orbitals we have previously developed.

The spin-orbit coupling plays an important role in Sr2IrO4. From our Kohn-Sham band structure,we estimate the value of the spin-orbit coupling constant ζSO as in the same order of magnitude as[43, 53, 161]: ζSO ≈ 0.4 eV. For the interested reader, the details of this calculation can be found inAppendix A. Since our previous effective model is from now on irrelevant to describe the t2g bands,a tight-binging approach in which iridium atomic levels are described in the the strong-spin-orbit-coupling limit is seducing.

In section 3.1, the impact of the corrections induced by the spin-orbit coupling on the eg andt2g states was thoroughly described and the TP-equivalence approximation was introduced. In thisapproach, the cubic crystal field is assumed large enough to treat the eg and t2g states still separatelyeven if the spin-orbit coupling is taken into account. As a result, the spin-orbit corrections in thet2g subspace imply to use a new basis which is composed of what we have called in section 3.1 the“jeff = 1/2” and the two “jeff = 3/2” states, whereas the spin-orbit interaction is ineffective on d3z2−r2and dx2−y2 states, since their orbital angular momentum is completely quenched. The TP-equivalenceapproximation appears particularly adapted to describe the formation of the electronic band structureof undistorted Sr2IrO4 with the spin-orbit coupling. Consequently, the t2g bands would come from thejeff = 1/2 and jeff = 3/2 atomic states.

Some details on the determination of the local jeff = 1/2 and jeff = 3/2 basis

The jeff = 1/2 and jeff = 3/2 states were introduced in section3.1 to describe the eigenstates ofa cubic system in which the spin-orbit coupling is taken into account within the framework of theTP-equivalence approximation. We remind here that their forms are the following:

|jeff =1

2,mj = −1

2〉 = 1√

3|dyz ↑〉 −

i√3|dxz ↑〉 −

1√3|dxy ↓〉

|jeff =1

2,mj = +

1

2〉 = 1√

3|dyz ↓〉+

i√3|dxz ↓〉+

1√3|dxy ↑〉

(3.27)


and:

|jeff =3

2,mj = −3

2〉 = 1√

2|dyz ↓〉 −

i√2|dxz ↓〉

|jeff =3

2,mj = +

3

2〉 = − 1√

2|dyz ↑〉 −

i√2|dxz ↑〉

|jeff =3

2,mj = −1

2〉 = 1√

6|dyz ↑〉 −

i√6|dxz ↑〉+

√2

3|dxy ↓〉

|jeff =3

2,mj = +

1

2〉 = − 1√

6|dyz ↓〉 −

i√6|dxz ↓〉+

√2

3|dxy ↑〉

. (3.28)

Following our previous discussion, the local basis composed of |d3z2−r2〉, |dx2−y2〉, |jeff = 1/2〉, |jeff =3/2, |mj | = 3/2〉, |jeff = 3/2, |mj | = 1/2〉 appears to be a natural choice to perform the calculation ofthe Wannier orbitals. Besides, this basis has a great advantage: the coefficients of each eigenstate isindependent of the values taken by the spin-orbit coupling constant ζSO and the cubic crystal field.

The program dmftproj was thus used with this local basis to describe the Ir 5d states in thesystem. Unfortunately, contrary to what have been expected, this basis does not diagonalize the localHamiltonian and leads to the appearance of several hybridization terms between these eigenstates.This occurs because the calculation performed in section 3.1 was made with two approximations:

• the TP-equivalence approximation, of course, which is not rigorously verified here.

• the degeneracy of the three t2g states.5

The calculation of the eigenstates of a system which takes into account the spin-orbit coupling and atetragonal splitting in the framework of the TP-equivalence approximation is presented in Appendix A.

5This last assumption does hold for our problem too. The small elongation of the IrO6 octahedra induces a tetragonalsplitting which puts the dxy higher in energy than the dxz and dyz. This is also the origin of the charge difference shownin table 5.3.

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2Energy (eV)

0

0.5

1

1.5

DO

S (s

tate

s.eV

-1)

Ir-d3z

2-r

2

Ir-dxy

Ir-dx

2-y

2

Ir-dxz

/dyz

Atomic state ChargeIr 5d d3z2−r2 0.013Ir 5d dx2−y2 0.033Ir 5d dxy 0.687Ir 5d dxz/dyz 0.737

Figure 5.12: Partial DOS associated to the Ir 5d eg and t2g characters for Sr2IrO4 in the undistortedsymmetry with spin-orbit coupling. The Fermi level is materialized by the dotted vertical line. Thecharge of each atomic state was calculated by integrating the corresponding DOS between −3.5 eVand the Fermi level.


It shows that the eigenvalues and the coefficients of the eigenvectors |jeff = 1/2〉, |jeff = 3/2, |mj | = 3/2〉and |jeff = 3/2, |mj | = 1/2〉 will explicitly depend in this case of the ratio η = 2Q1/ζSO. Since noestimation of this ratio can easily be calculated, another approach was used to find the local ba-sis which described at best our local problem: it consists in evaluating numerically the basis whichdiagonalizes the density matrix in the considered energy window for calculating the Wannier projectors.

More precisely, the method consists in calculating a first time the Wannier orbitals based on thestandard (complex or cubic) basis in order to obtain the density matrix associated to them. And then,the basis which diagonalizes this density matrix is found and used in a second run to calculate directlythe Wannier orbitals. In this approach one must use a large energy window which contains all thecomplex (or cubic) orbitals initially. In the case of undistorted Sr2IrO4 in which the spin-orbit cou-pling is included, the energy window is [−3.5; 6.5] eV. The Wannier orbitals finally obtained with thismethod diagonalizes the density matrix at the end of the program dmftproj, which is also of practicaladvantage for a CTQMC calculation because the sign problem in CTQMC arises mostly because ofoff-diagonal terms in the local Hamiltonian.

Our approach can of course be improved further. In the current framework, the integration overthe Brillouin zone used in dmftproj and in the CTQMC calculations are not the same (a tetragonalintegration for the former and a point integration for the latter). Because of this difference, even ifthe Wannier projectors diagonalize the density matrix at the end of dmftproj, some small off-diagonalterms (of maximal order of magnitude 0.001 eV) will appear in the local Hamiltonian used in theCTQMC calculations. They can easily be neglected during the calculations but we have no control onthese terms yet. Another improvement would be achieved by restricting the size of the energy windowor restrict the projection to a subspace of orbitals (only the t2g for instance).

Introduction of the “jeff = 1/2” and “jeff = 3/2 bands”

With the previously described method, the best basis for our local problem was evaluated to be thefollowing vectors6:

|ψ1〉 = +0.99429 |d3z2−r2 ↓〉 +0.07543 |dxz ↑〉 +i0.07543 |dyz ↑〉|ψ1〉 ≈ |d3z2−r2 ↓〉

|ψ2〉 = +0.99018 |dx2−y2 ↓〉 −i0.10487 |dxy ↓〉 −0.06536 |dxz ↑〉 −i0.06536 |dyz ↑〉|ψ2〉 ≈ |dx2−y2 ↓〉

(5.3)

The deviations of this two first states from d3z2−r2 and dx2−y2 respectively can be understood as ameasure of the validity of the TP-equivalence approximation. The numerical method used does indeedtake into account the off-diagonal terms due to the spin-orbit coupling corrections, which are neglectedby using this approximation. Looking at the value of the coefficients, a tight-binding model based oniridium described in the TP-equivalence approximation framework is thus a very good approach.

6We display here only half of the basis. The results are similar for the states |d3z2−r2 ↑〉, |dx2−y2 ↑〉, |jeff = 1/2,mj =+1/2〉, |jeff = 3/2,mj = +1/2〉 and |jeff = 3/2,mj = −3/2〉.


|ψ3〉 = −i0.00588 |dx2−y2 ↓〉 −0.62942 |dxy ↓〉 −i0.54945 |dxz ↑〉 +0.54945 |dyz ↑〉|ψ3〉 = |jeff = 1

2 ,mj = −12〉

|ψ4〉 = −i0.13967 |dx2−y2 ↓〉 +0.76996 |dxy ↓〉 −i0.44026 |dxz ↑〉 +0.44026 |dyz ↑〉|ψ4〉 = |jeff = 3

2 ,mj = −12〉

|ψ5〉 = +i0.10668 |d3z2−r2 ↓〉 −i0.70307 |dxz ↑〉 −0.70307 |dyz ↑〉|ψ5〉 = |jeff = 3

2 ,mj = +32〉

(5.4)

The three vectors |ψ3〉, |ψ4〉 and |ψ5〉 are thus associated to the t2g states. The first column of coeffi-cients comes from the deviations from the TP-equivalence approximation and can thus be neglected.The three last column of coefficients of these vectors are really close to the “idealized” case of theexpressions (3.27) and (3.28).7

The state |ψ5〉 is thus the state |jeff = 3/2,mj = +3/2〉. The discrepancy of the value of thecoefficients of |ψ3〉 and |ψ4〉 from the theoretical “idealized” case comes from the non-degeneracy ofthe t2g orbitals because of the tetragonal crystal-field splitting Q1. Using the expressions calculatedin Appendix A, the value of the ratio η = 2Q1/ζSO is positive and estimated to be about 0.400. Thisimplies that the local dxy states has an higher energy than the local dxz and dyz states, which is ingood agreement with an elongation of the IrO6 octahedron along the z-axis. By using the value ofthe spin-orbit coupling in the compound – ζSO ≈ 0.4 eV–, the tetragonal splitting is found to beQ1 = 0.08 eV. The fitting between the numerical coefficients and the theory confirms that we can labelthese states as |jeff = 1/2,mj = −1/2〉 and |jeff = 3/2,mj = −1/2〉.

As a result, the Kohn-Sham band structure of undistorted Sr2IrO4 with the spin-orbit couplingseems to be explained by a tight-binding approach in which only iridium sites are considered and withon-site atomic levels described by the local basis made of the states d3z2−r2 dx2−y2 jeff = 1/2 andjeff = 3/2. At the beginning of this study, it was highlighted that each band is twice degenerated.Whereas for d3z2−r2 and dx2−y2 the spin is still a good quantum number, this degeneracy can beassociated to the mj number associated to jeff = 1/2 states – mj = ±1/2 – and jeff = 3/2 states –mj = ±1/2 and mj = ±3/2.

To confirm the validity of our approach, the magnitude of each character along the Kohn-Shamband structure is plotted in figures 5.13 and 5.14. Whereas the jeff = 1/2 state is mainly associated toonly one band (the “upper” t2g band), the two jeff = 3/2 states are mixed along the two other bands.Moreover close to the Γ point, the jeff = 1/2 and the jeff = 3/2 |mj | = 1/2 characters are mixed. Thismixing of several characters along each band was not expected since the local basis is orthonormal anddiagonalizes the density matrix. That is why we assume that these mixings might come from somenumerical errors: the subprogram we use to plot thes characters indeed performs a point integrationinstead of the integration based on the tetragonal weight as in dmftproj. This might introduce somehybridization between the local states. A more detailed study must be carried out to confirm this idea.

Consequently, in a first approach, we will consider that a one-to-one correspondence between thebands and the jeff = 1/2 and jeff = 3/2 states is possible:

• The “upper” t2g band (in light green on the left picture of figure 5.13) which crosses the Fermilevel can be understood as the “jeff = 1/2 band ”.

7We remind to the reader the following numerical values:√

1/6 ≈0.40824,√

1/3 ≈0.57735,√

1/2 ≈0.70710 and√2/3 ≈0.81649.


Γ M X Γ

-3

-2

-1

0

1

2

Ene

rgy(

eV)

0

0.2

0.4

0.6

0.8

1

Γ M X Γ

-3

-2

-1

0

1

2

Ene

rgy

(eV

)

Figure 5.13: Magnitude of the Wannier character named jeff = 1/2 along the bands in the energywindow [−3.5; 2] eV. The scale in renormalized arbitrary unit gives the weight of the Wannier characterin the band. The band structure of undistorted Sr2IrO4 with the spin-orbit coupling is reminded onthe left picture.

0

0.2

0.4

0.6

0.8

1

Γ M X Γ

-3

-2

-1

0

1

2

Ene

rgy

(eV

)

0

0.2

0.4

0.6

0.8

1

Γ M X Γ

-3

-2

-1

0

1

2

Ene

rgy

(eV

)

Figure 5.14: Same as the right picture above. Left picture corresponds to the character jeff = 3/2|mj | = 1/2, right picture to the character jeff = 3/2 |mj | = 3/2


-3 -2 -1 0 1 2Energy (eV)

0

0.5

1

1.5

DO

S (s

tate

s.eV

-1)

d3z

2-r

2

dx

2-y

2

jeff

=1/2

jeff

=3/2 - |mjeff

|=1/2

jeff

=3/2 - |mjeff

|=3/2

Wannier orbital Charged3z2−r2 0.010dx2−y2 0.030jeff = 1/2 1.147jeff = 3/2 |mj | = 1/2 1.820jeff = 3/2 |mj | = 3/2 1.826

Figure 5.15: Partial DOS associated to each Wannier character for Sr2IrO4 in the undistorted symmetrywith spin-orbit coupling. The table on the right presents the charge repartition between the Wannierstates.

• the “middle” t2g band (in light blue) is associated to the jeff = 3/2 |mj | = 3/2 state.

• the “lower” t2g band (in purple) can be understood as the “jeff = 3/2 |mj | = 1/2 band ”.

The filling of each band can thus be deduced from the integration of the DOS of the correspondingWannier states up to the Fermi level. The partial DOS associated to each Wannier orbital is displayedon the left part of figure 5.15 and the corresponding table presents the results of the integration: thejeff = 1/2 band is a little more than half-filled, whereas the two jeff = 3/2 bands are almost filled andmoreover equally filled.

To conclude, this study has highlighted that the band structure of undistorted Sr2IrO4 with thespin-orbit coupling can be easily understood if the jeff = 1/2 and jeff = 3/2 states local basis is used:the jeff = 1/2 band is then confined almost alone close to the Fermi level and is almost half-filled,whereas the jeff = 3/2 are almost completely filled.

This new distribution of the bands will change also drastically the influence of the correlationson the system. In a LDA+DMFT calculation, it is expected that the jeff = 3/2 bands will be firstlycompletely full, reducing then the problem to the metal-insulator transition of one half-filled band. Theeffective dimensionality of the system is then reduced, which eases a lot the impact of the correlations.

However, the bandwidth of the jeff = 1/2 band is of about 1.8 eV, which implies that the criticalvalue for the Coulomb repulsion parameter U would lie between 2.2 and 2.7 eV in a pure one-bandmodel. Since some charge redistribution between the jeff = 1/2 and jeff = 3/2 bands is expected,these values must be increased again. A Coulomb parameter U of 2 eV maximum will thus lead to acorrelated metallic state only, although the metal-insulator transition is not so far. As a result, thespin-orbit coupling helps to reinforce the power of the electronic correlations by reducing the effectivedimensionality of the problem and we will see in the following that the distortions will enhance evenmore this phenomenon.


5.1.3 Case 3: Modifications in the electronic band structure induced by the dis-tortions

Structural properties

As we have already explained, according to X-ray and neutron diffraction studies [41, 72, 88, 135, 156],Sr2IrO4 is assigned the tetragonal I41/acd space group. It corresponds to the symmetry of a K2NiF4-type compound in which the corner-shared octahedra are not well-aligned along the crystallographicdirection a and b, but are alternately rotated clockwise and anticlockwise around the c-axis by about11. The corresponding space-group is usually called I41/acd but for the sake of simplicity, we willrefer to it as the “distorted ” symmetry in the following. It results in the formation of a superstructureof size

√2at×

√2at× 2ct – where at and ct are the parameter of the corresponding tetragonal K2NiF4

structure – and which contains four formula units.

The conventional representation of the unit cell of Sr2IrO4 in this distorted symmetry is depicted infigure 5.16. The compound is still a stacking of Ir-O2 and Sr-O planes. The lattice constants we usedwere: a = 5.497 Å (at = 3.887 Å) and c = 25.798 Å (ct = 12.899 Å) (taken from the values obtainedat 295 K by Crawford et al. [41]). In the following, the c-axis and the z-axis are the same, but thex-axis and y-axis are turned of 45 from the crystallographic directions a and b, so that they wouldstill point to the corner of the IrO6 octahedra if they were no distortions. The detail of the atomicpositions can be found in the first column of table 5.4.

Atomic species Position Muffin-tin radiusin reduced coordinates in Bohr radius a0 = 0.529Å

Sr (0,0.25,0.550) 2.31 a0Ir (0,0.25,0.375) 1.97 a0O1 (0.201,0.451,0.125) 1.75 a0O2 (0,0.25,0.45475) 1.75 a0

Table 5.4: Description of the structural parameters used for our DFT calculation on Sr2IrO4 in thedistorted I41/acd symmetry.

The distortions do not change the electronic state of each atomic species in the crystal which stillare Sr2+, O2− and Ir5+. Iridium atoms still accommodate 5 valence electrons in their 5d orbitals.The corner-shared IrO6 octahedra are not well-aligned along x and y directions ( IrO1Ir = 157.8).Moreover the octahedra are elongated along the c axis: Ir-O1=1.980 Å and Ir-O2=2.057 Å (+3.8%).It induces an additional tetragonal field, which leads to a destabilization in energy of the dxy orbitalin comparison to the dxz and dyz states. Thus the chemical picture predicts that 1 electron remains inthe dxy level. However, we will see that the distortions will introduce another effect which cancels thisphenomenon and leads to a configuration different from this molecular picture. Be it as it may, the5 valence electrons will thus occupy the three degenerate t2g orbitals (dxy, dxz and dyz) and the twodegenerate eg orbitals (d3z2−r2 and dx2−y2), with higher local energy, will be empty. From this result,the band theory predicts a metallic description for distorted Sr2IrO4.

Technicalities of the DFT calculation

We have performed our DFT calculation on this distorted description of Sr2IrO4 with the same pa-rameters as for the undistorted case: RMT .Kmax = 7, lmax = 10 and lns,max = 4. The local densityapproximation (LDA) was used for the treatment of the exchange-correlation term, and the irreducibleBrillouin zone was sampled with 99 k-points (using a tetrahedral mesh).


Figure 5.16: Conventional unit cell of Sr2IrO4 in the “distorted ” structure (I41/acd symmetry). Thegreen spheres stand for the strontium ions (Sr), the golden ones for Iridium (Ir) and the red ones forOxygen (O). The corner-shared IrO6 octahedra are alternately rotated clockwise and anticlockwisearound the c-axis by about 11. From [88]

Figure 5.17: Conventional Brillouin zone for Sr2IrO4 in the distorted symmetry. The high symmetrypoints are Γ(0, 0, 0), Z(0, 0, 1), M(0.5, 0, 0), X(0.5, 0.5, 0), P (0.5, 0.5, 0.5) and N(0.5, 0, 0.5). The greenarrows depict the k-path which will be used to represent the electronic band structure in the following.The upper part depicts the correspondence between the distorted Brillouin zone (in red with orangeaxes) and the undistorted Brillouin zone (in black with blue axes).


The muffin-tin radii chosen for each atomic species are presented in the second column of table 5.4.They are different from the previously used values of table 5.1 because the lattice parameters (at andct) are not exactly the same as those of the undistorted structure studied in the previous sections. Thiswill create some little changes in the calculation of the partial DOS or in the bandwidth of the bands,but the main physical phenomena will not depend on this.

The energy threshold was set at −7.5 Ry. The categorization of the electronic states of each atomicspecies described in table 5.2 still holds. The total semi-core and valence states considered during thecalculation was then of 46+29=75 electrons times 4 – because there are now 4 formula units in theunit cell–, that is to say 300 electrons.

The Kohn-Sham band structure was plotted along the k-path depicted in figure 5.17. In fact, theconventional Brillouin zone for the distorted structure has the same shape than the undistorted one butsince the unit cell was quadrupled in the distorted symmetry, the Brillouin zone is in fact four timessmaller than previously. It is however possible to find a correspondence between these two Brillouinzones, as depicted on figure 5.17. For instance in the kz = 0 plane, because of the

√2×

√2 superlattice:

• theX and Γ points of the undistorted Brillouin zone become equivalent in the distorted symmetry,

• theM point of the undistorted Brillouin zone corresponds to theX point of the distorted Brillouinzone,

• the M point of the distorted Brillouin zone is located in the middle of the segment [ΓX] of theundistorted Brillouin zone.

These relations will enable us to compare more easily the Kohn-Sham band structure obtained fordistorted Sr2IrO4 with its undistorted counterpart, and thus to shed the light on the effects inducedby the structural distortions.

Kohn-Sham band structure of distorted Sr2IrO4

As expected, our LDA calculation without spin-orbit coupling predicts a metallic nature for Sr2IrO4

in the distorted symmetry. As observed on figure 5.18, the total DOS has indeed a finite value at theFermi level but a separation in energy is obvious between 0.4 and 1.3 eV. The set of bands spreads lesslow in energy. However the same coarse band ordering can be found when a study relying on a tightbinding approach is performed:

• Between −8.5 eV and about −4.5 eV, lie the oxygen bands which come from the bonding eg (σ)and t2g (π) molecular orbital between the Ir 5d and the O 2p states and have mostly an O 2pcharacter.

• Between −4.5 eV and −2 eV, one finds the oxygen bands which come from the non-bondingoxygen molecular orbitals.

• From −2 eV, lie the eg and t2g bands.

In the following, we will again focus our attention to the eg and t2g bands only, since they still dominatethe physical properties of the material.

In the Kohn-Sham band structure depicted on figure 5.19, the number of bands is four timesbigger than in the previous undistorted case, because there are now four formula units in the unitcell. As a result, 12 t2g bands can be counted (4 blue and 8 green) between −2 and 0.4 eV. Up to 6bands cross the Fermi level: on the [ΓM ] segment, four green bands and two blue bands cross indeed it.


-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2Energy (eV)

0

2

4

6

8

10

12

14

DO

S (s

tate

s/eV

-1)

TotalIrSrO

1

O2

Figure 5.18: Total and partial DOS of Sr2IrO4 in the distorted symmetry without spin-orbit coupling.The Fermi level is materialized by the dotted vertical line. Each curve was divided by four (thusdescribing only one formula unit) in order to ease the comparison with the undistorted case.

Z Γ M X Γ P N -3

-2

-1

0

1

2

Ene

rgy(

eV) E

FE

FE

FE

F

Figure 5.19: Kohn-Sham band structure of Sr2IrO4 in the distorted symmetry without the spin-orbitcoupling. The eg bands (d3z2−r2 in yellow and dx2−y2 in red) are well-separated from the t2g bands (inblue and green on the figure). The size of the unit cell was quadrupled, implying a multiplication by4 of the number of bands in comparison with the undistorted case. There is a mirror effect for the dxzand dyz bands.


The partial DOS associated to the Ir 5d characters are depicted on figure 5.20. The separation inenergy lies between a lower set of bands whose characters are essentially of t2g-type and an upper setof bands whose characters are mainly of eg-type. By plotting the Ir 5d character along the Kohn-Shamband structure, the green and yellow bands in figure 5.19 has a pure character: dxz and dyz for theformer and d3z2−r2 for the latter. On the contrary, dxy and dx2−y2 characters are mixed along theblue and red bands. However, the dxy contribution is more important along the blue one and dx2−y2character is the most present along the red one. The structure of the partial DOS associated to thesestates in figure 5.20 confirms also this picture. As a result, we will refer to the blue (red) band as the“dxy band ” (“dx2−y2 band ” respectively) in the following.

In order to compare the Kohn-Sham band structures of undistorted and distorted Sr2IrO4, a similarsupercell of size

√2a×

√2a×2c was constructed for undistorted Sr2IrO4 and the corresponding Kohn-

Sham band structure is depicted in figure 5.21-(b) along the same k-path we used for the distortedcase. The correspondence between the undistorted and the distorted Brillouin zones was explainedpreviously and enables to understand how the figure 5.21-(b) can be obtained from figure 5.8:

• By “folding along M ” in figure 5.8, the bands along the [ΓM ] segment and those along the [M X]segment are superimposed, which gives the band structure observed along the [Γ X] segment infigure 5.21-(b).

• Similarly, by “folding along the middle of the [Γ X] segment” in figure 5.8, the Kohn-Sham bandstructure observed along the [Γ M ] segment in figure 5.21-b is found.

• Each band of the obtained band structures must of course be “twice drawn” to take into accountthe doubling of the size along the c-axis in the supercell.

As a result, the folding of the bands result in a “mirror effect ” for the dxz and dyz (green) bands andthe dxy (blue) band. Moreover, the dx2−y2 (red) bands has now an overlap with the t2g bands, as it canbe seen at the Γ point of figure 5.21-(b). When the distortions are taken into account, the symmetryof the system is lowered and the dxy and dx2−y2 bands are no longer orthogonal. The hybridizationbetween the two leads then to the formation of a gap between a bonding and an antibonding bands,which corresponds actually to the red and the blue bands observed on figure 5.21-(a). The same phe-nomenon does not occur between the dx2−y2 band and the dxz, dyz bands because their orthogonalityis preserved despite the distortions. As a result, the dxz, dyz and d3z2−r2 bands are not affected muchby the distortions, as shown by comparing figures 5.21-(a) and (b).

By comparing the two pictures, a narrowing of the dxz and dyz bands can be also noticed (from[−1.35; 0.27] eV to [−1.09; 0.36] eV). Because of the rotations of the IrO6 octahedra, the hopping be-tween Ir and O1 sites decreases, which leads to this reduction of the bandwidth. However, this effectis strongly increased between figures 5.21-(a) and (b) because of a small default in our modelization8.The bandwidth of the dxz and dyz bands measured in figure 5.8 is indeed [−1.11; 0.39] eV. The reduc-tion of the bandwidth due to the distortions is thus more of an order of a few percent.

By creating the bonding and the antibonding bands with the dx2−y2 band, the dxy band was effec-tively pushed lower in energy and thus more filled. By calculating the charge associated to the Ir 5dstates – obtained by integrating the corresponding partial DOS up to the Fermi level – in table 5.5,this effect appears clearly: whereas in table 5.3, the dxy state has slightly less charge than the dxz anddyz states, the former is now more filled than the latter, although the elongation of the IrO6 octahedraalong the c-axis is more pronounced in the undistorted structure we used.

8While creating the undistorted supercell, we have reduced the size of the IrO6 octahedra by accident. The hybridiza-tion between Ir and O1 was then overestimated during the calculation, hence the wider bands.


-3 -2 -1 0 1 20

0.5

1

1.5

2

2.5

3

DO

S (s

tate

s.eV

-1)

Ir-d3z

2-r

2

Ir-dx

2-y

2

Ir-dxy

Ir-dxz

/dyz

Figure 5.20: Partial DOS of the Iridium eg and t2g characters for Sr2IrO4 in the “undistorted ” symmetrywithout spin-orbit coupling. The energy are given with respect to the Fermi level, materialized by thedotted vertical line.

Z Γ M X Γ P N (a) - with distortions

-3

-2

-1

0

1

2

Ene

rgy(

eV) E

FE

FE

FE

F

Z Γ M X Γ P N (b) - without distortions

-3

-2

-1

0

1

2

Ene

rgy(

eV) E

F

Figure 5.21: Comparison of the Kohn-Sham band structures between distorted Sr2IrO4 (panel a) andundistorted Sr2IrO4 (panel b) without the spin-orbit coupling. Both band structures are representedalong the same k-path (in the distorted I41/acd symmetry). The hybridization between the dx2−y2and dxy bands (in red and blue respectively) leads to a the formation of the energy gap between 0.4 eVand 1.3 eV.


Construction of a localized eg-t2g basis set Wannier-type basis set for distorted Sr2IrO4

In order to get a better idea of the filling of the t2g bands, the Wannier orbitals associated to thesebands have been calculated. We have built the Wannier projectors from the cubic basis in the energywindow [−2.8; 0.4] eV. The eg states were omitted on purpose in our approach, since their charge isalmost zero in the considered energy range. The hybridization between dxy and dx2−y2 bands wereconsidered also small enough to be neglected (0.146 eV in the chosen energy window).

The obtained results are in good agreement with figure 5.19: the Wannier functions built from thedxz and dyz states indeed correspond to the green bands and the Wannier function built from the dxystates describes exactly the blue band. In fact, no information in the description of the antibonding dxyband was lost in the calculation, thanks to the orthogonalization process of the Wannier functions. Thefilling of each band is presented in the second column of table 5.5. The dxy band is almost completelyfilled, as expected. It thus remains almost three electrons in the dxz and dyzbands.

Charge calculated from of the Wannier functionsthe partial DOS of the energy window [−2.8; 0.4] eV

Ir 5d d3z2−r2 0.027 0.027Ir 5d dx2−y2 0.096 0.097Ir 5d dxy 0.841 1.936Ir 5d dxz /dyz 0.672 1.540

Table 5.5: Charge repartition between the eg and t2g atomic states (first column) and the eg and t2gbands (second column), between −2.8 eV and the Fermi level. The Wannier orbitals were obtainedfrom a calculation based on the energy window [−2.8; 0.4] eV.

To conclude, by taking into account the rotations of the IrO6 octahedra around the c-axis, thedescription of Sr2IrO4 requires to use four formula units in the less symmetric unit cell. It impliesthat each undistorted band is folded four times in the obtained Kohn-Sham band structure, whichintroduces an hybridization between the dxy and the dx2−y2 bands. As a result, an antibonding anda bonding bands are formed and a gap is opened between 0.4 and 1.3 eV. Besides, the bonding bandcan be considered as completely filled.

The electronic correlations will thus mostly affect the dxz and dyz bands only. In an LDA+DMFTapproach, the Hubbard model used for the calculation will reduce to two 3/4-filled bands. It is thusexpected that an insulating state will be more easily reached than in the undistorted case since thedimensionality of the problem has decreased and also because the bandwidth of the dxz and dyz bands– about 1.45 eV – is much narrower than the one of the dxy band in the undistorted case. However,with a Coulomb parameter of 2 eV maximum, it seems difficult to obtain an insulating state for thissystem.

5.1.4 Influence of the spin-orbit coupling and the distortions together

As we have already mentioned in the introduction, LDA calculations predict a metallic state for Sr2IrO4

and four bands cross the Fermi level (cf. figure 5.1). In this part, we use the previous studies – on thespin-orbit coupling and the distortions – to understand the Kohn-Sham band structure of Sr2IrO4 andabove all explain the nature of these four bands.

To obtain the DFT description of Sr2IrO4, the spin-orbit corrections were introduced on the iridiumatoms of the previous distorted structure. Since the calculation is not spin-polarized, the Kohn-Sham


band structure can be plotted along the same k-path (in the same Brillouin zone) as in part 5.1.3, andthe correspondence between the distorted and the undistorted Brillouin zone can be used again.

The Kohn-Sham band structure of Sr2IrO4

The method used for the calculation suggests first to draw a comparison between the Kohn-Sham bandstructures of distorted Sr2IrO4 with and without spin-orbit coupling. Both of them are depicted infigure 5.22. It appears clearly that the spin-orbit corrections do not affect the eg bands (the d3z2−r2band represented in red and the dx2−y2 band in yellow on both pictures), whereas deep modificationsoccur in the set of t2g bands. The comparison of the corresponding total DOS – both represented infigure 5.23 – confirms of course this conclusion. Moreover, it highlights that the energy separationbetween the eg and the t2g bands has a little decreased: from [0.4; 1.3] eV to [0.5; 1.27] eV.

Consequently, a similar approach based on the TP-equivalence approximation appears to be validalso in the distorted case. However, as explained in part 5.1.3, the red and blue bands in figure 5.22 (b),are the bonding and antibonding bands which results from the hybridization of the initial dxy and dx2−y2bands and thus mix these two Ir 5d characters. As a result, a modified version of the TP-equivalenceapproxmation introduced in section 3.1 must be considered strictly speaking. We will call it in thefollowing the “distorted TP equivalence approximation”. This version must consider the two linearcombinations of dxy and dx2−y2 respectively associated to the antibonding and the bonding states.The former will be unaffected by the spin-orbit coupling as the d3z2−r2 state, and the latter will beinvolved in the formation of the jeff = 1/2 and the jeff = 3/2 states. It can therefore be expected thatthe corresponding jeff = 1/2 and the jeff = 3/2 bands will differ from the previous jeff = 1/2 and thejeff = 3/2 bands introduced in the undistorted case with spin-orbit coupling.

To better understand the impact of the distortions on the jeff = 1/2 and jeff = 3/2 bands, it isalso possible to imagine that the distortions were introduced in the undistorted structure “once” thespin-orbit coupling was already included. In this case, the supercell of size

√2a ×

√2a × 2c can be

considered again for undistorted Sr2IrO4 in order to make the comparison easier between the two bandstructures. The result is displayed in figure 5.24 (b). The color code used for the bands is the same asin the section 5.1.2:

• the light green bands are the jeff = 1/2 bands and their replicas,

• the light blue bands are the jeff = 3/2 |mj | = 3/2 bands,

• the purple bands are the jeff = 3/2 |mj | = 1/2 bands.

The folding method – described in part 5.1.3 – can also be applied to understand how the figure 5.24-(b)can be obtained from the figure 5.11. The main consequence of these foldings is the overlap betweenthe dx2−y2 (red) bands and the jeff = 1/2 and jeff = 3/2 bands, as it can be seen at the Γ point of fig-ure 5.24-(b). As we have explained in the previous subsection, the dxy and dx2−y2 bands are no longerorthogonal when the structural distortions are introduced. In a picture where the spin-orbit couplingare taken into account, this implies that both the jeff = 1/2 and jeff = 3/2 |mj | = 1/2 bands arenot orthogonal anymore with the dx2−y2 bands. Therefore, the formation of bonding and antibondingbands involves now these three types of band. It is however expected that the jeff = 1/2 and jeff = 3/2|mj | = 1/2 states are not mixed during this process since they still are orthogonal. Moreover, thejeff = 3/2 |mj | = 3/2 band must not be modified much by the distortions.

By comparing this analysis with the electronic band structure of figure 5.24-(a), more changes havehappened than expected, but the main features previously described allow us to say that:


Z Γ M X Γ P N (a) - with Spin-orbit

-3

-2

-1

0

1

2

Ene

rgy(

eV) E

FE

FE

FE

F

Z Γ M X Γ P N (b) - without Spin-orbit

-3

-2

-1

0

1

2

Ene

rgy(

eV)

EF

EF

EF

EF

Figure 5.22: Comparison of the Kohn-Sham band structure of distorted Sr2IrO4, with (panel a) andwithout (panel b) the spin-orbit coupling. The d3z2−r2 band (in yellow) and the dx2−y2 band (in red)are not affected by the spin-orbit corrections. On the contrary, the t2g bands are modified a lot.

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2Energy (eV)

0

2

4

6

8

10

12

14

DO

S (s

tate

s.eV

-1)

without Spin-Orbitwith Spin-Orbit corrections

Figure 5.23: Total DOS for Sr2IrO4 in the distorted symmetry. The red curve includes the correctionsinduced by taking into account the spin-orbit coupling. The Fermi level is materialized by the dottedvertical line.


Z Γ M X Γ P N (a) - with distortions

-3

-2

-1

0

1

2

Ene

rgy(

eV) E

FE

FE

FE

F

Z Γ M X Γ P N (b) - without distortions

-3

-2

-1

0

1

2

Ene

rgy(

eV)

EF

EF

EF

EF

Figure 5.24: Comparison of the Kohn-Sham band structure between distorted Sr2IrO4 (panel a) andundistorted Sr2IrO4 (panel b) with the spin-orbit coupling. Both band structures are represented alongthe same k-path (in the distorted I41/acd symmetry). The hybridization between the dx2−y2 and dxybands (in red and blue respectively) leads to a the formation of the energy gap between 0.4 eV and1.3 eV.

• at the Γ point, the “upper” (light brown) band must derive from the hybridization between thedx2−y2 and the jeff = 3/2 |mj | = 1/2 bands.

• the four bands close to the Fermi level (in dark brown on the figure) certainly derive only fromthe jeff = 1/2 and jeff = 3/2 |mj | = 3/2 bands: the upper intersection at the Γ point clearlycomes from the previous jeff = 1/2 bands, whereas the lower intersection can be understood asthe one from the jeff = 3/2 |mj | = 3/2 band.

However, in order to confirm these statements, a study based on the Wannier functions was carriedout. Its results are discussed in the next paragraph.

Nature of the four bands crossing the Fermi level in Sr2IrO4

As explained previously, the local Ir 5d basis which describes at best Sr2IrO4 must rely on a modified TPequivalence approximation framework, where the hybridization between the dxy and the dx2−y2 statesinduced by the structural distortions is taken into account. For the sake of simplicity, we decidedto evaluate it numerically by using the same approach which was presented in part 5.1.2. We usedthe energy window [−4.0; 6.8] eV in order to have a rather complete description of the antibondingand bonding bands which come from the dxy and dx2−y2 bands. The obtained basis vectors were thefollowing:9

9Here again, we will study only half of the basis. The results are similar for the other half.


|d3z2−r2 ↓〉 |dx2−y2 ↓〉 |dxy ↓〉 |dxz ↑〉 |dyz ↑〉|ψ1〉 0.9926 0 0 0.0862 0.0862|ψ2〉 0 0.9525 0.2865 0.0724 0.0724

|ψ3〉 0 0.18303 0.46377 0.61295 0.61295|ψ4〉 0 0.24310 0.83833 0.34503 0.34503|ψ5〉 0.12196 0 0 0.70183 0.70183

Table 5.6: Modulus of the coefficients of each states (|ψi〉)i=1,5 on the cubic orbitals.

|ψ1〉 = +0.9571 + i0.2629 |φd3z2−r2↓〉 +0.0853 + i0.0125 |φdxz↑〉 +0.0125− i0.0853 |φdyz↑〉

|ψ2〉 = +0.9085 + i0.2863 |φdx2−y2↓〉 −0.2863 + i0.0121 |φdxy↓〉

+0.0397 + i0.0605 |φdxz↑〉 −0.0605 + i0.0397 |φdyz↑〉(5.5)

These two states have most of their weight on the eg orbitals. On the contrary, the three othersare mainly decomposed on the t2g levels:

|ψ3〉 = −0.04023 + i0.17855 |φdx2−y2↓〉 −0.17855 + i0.42802 |φdxy↓〉

+0.03950− i0.61168 |φdxz↑〉 +0.61168 + i0.03950 |φdyz↑〉

|ψ4〉 = +0.16869− i0.17504 |φdx2−y2↓〉 +0.17504− i0.81985 |φdxy↓〉

−0.01730− i0.34460 |φdxz↑〉 +0.34460− i0.01730 |φdyz↑〉

|ψ5〉 = −0.00874 + i0.12165 |φd3z2−r2↓〉 −0.03558− i0.70093 |φdxz↑〉 −0.70093 + i0.03558 |φdyz↑〉

(5.6)Contrary to the results obtained in part 5.1.2, the coefficients of these vectors in the cubic basis

are complex – with real and imaginary parts –. In order to go further in the study, we will focus ourattention on the modulus associated to each coefficient10 and support our analysis by a comparisonbetween the values of table 5.6 with the theoretical coefficients of the states obtained in the true TPequivalence approximation framework (cf. expressions (3.27) and (3.28)).

The contribution of the dxz and the dyz states in the vectors |ψ1〉 and |ψ2〉 can be easily neglected.In this case, |ψ1〉 obviously corresponds to the d3z2−r2 state and |ψ2〉 is the linear combination of thestates dx2−y2 and dxy with a major weight on the dx2−y2 orbital. This vector can then be seen asdescribing the antibonding level associated to the (red) dx2−y2 band on figure 5.22 (a).

Moreover, by neglecting the contribution of the d3z2−r2 orbital in |ψ5〉, the decomposition of thisbasis vector becomes really close to the state |jeff = 3/2,mj = +3/2〉 (whose coefficients are equal to√1/2 ≈ 0.70710). As discussed previously, no great change was expected for this level and we will

then consider in the following that |ψ5〉 describes the |jeff = 32 ,mj = +3/2〉 state in the local basis.

The last vectors |ψ3〉, |ψ4〉 must then be linked to the “modified” |jeff = 1/2,mj = −1/2〉 and|jeff = 3/2,mj = −1/2〉 states, which involve the bonding linear combination between dx2−y2 and dxy.If we deliberately omit the coefficient before the dx2−y2 orbital, the decomposition in modulus of |ψ3〉

10Some works are in progress to understand whether the phase factors have a physical meaning or whether they arisefrom pure numerical reasons.


reminds the expression of |jeff = 1/2,mj = −1/2〉, whereas the decomposition of |ψ4〉 seems quite closeof the expression of |jeff = 3/2,mj = 1/2〉. 11 This argument tends to show that |ψ3〉 (|ψ4〉) can beunderstood as |jeff = 1/2,mj = −12〉 (|jeff = 3/2,mj = −1/2〉 respectively).

To confirm this idea, the magnitude of the Wannier characters |ψ3〉, |ψ4〉 and |ψ5〉 were plottedalong the Kohn-Sham band structure. The results are displayed in figures 5.25 and 5.26. As it canbe seen on the right picture of figure 5.25, the upper intersection with the Γ point at 0.5 eV can beattributed to the |ψ3〉 character. Furthermore, the two “legs” on the segment [Γ X] and [Γ M ] between−1.5 and −3 eV have mostly a |ψ4〉 character on the left picture of figure 5.26. These two featuresrespectively belong to the jeff = 1/2 and the jeff = 3/2 mj = −1/2 bands, as discussed previously.With this last argument, we thus state that:

• the |ψ3〉 Wannier character is the “modified ” |jeff = 1/2,mj = −1/2〉 of the model which describedthe local problem in the “distorted TP equivalence approximation” framework.

• the |ψ4〉 Wannier character is the “modified ” |jeff = 3/2,mj = −1/2〉 of the same model.

With these results in mind, the four bands crossing the Fermi level in Sr2IrO4 are mainly de-scribed by the “modified ” jeff = 1/2 states but also by the jeff = 3/2 |mj | = 3/2 states, as expectedfrom our previous discussion. The intersection with the Γ point at 0.05 eV is doubtless of character |ψ5〉.

However, the energy window we used to find this local basis and the corresponding Wannier func-tions is obviously too large. Consequently, the basis is too localized in the real space and that iswhy the bands mixes several character along the considered k-path. A new calculation with a smallerenergy-range – which typically contains only the four bands crossing the Fermi level – must be per-formed to find the Wannier functions associated to them. It is strongly expected that only one Wanniercharacter is necessary to describe them: these four bands can indeed be seen as the foldings of only oneband whose character is a linear combination of the “modified ” jeff = 1/2 and the jeff = 3/2 |mj | = 3/2states. Nevertheless, this requires to improve the method used to find the local basis on which theWannier functions are built. This work is still currently in progress but this study already shows thatthe four band crossing the Fermi level cannot be understood as “pure jeff = 1/2 states”, contrary towhat is commonly taught in the literature [84].

Although the true Wannier function which describes these four bands is not known yet, it is possibleto anticipate the effect of the electronic correlations in Sr2IrO4. The four bands close to the Fermi levelare indeed the foldings of a band whose width can be estimated as about 1 eV. Moreover, this bandis half-filled as can be confirmed by integrating the total DOS between the Fermi level and −0.5 eV.Consequently, in an LDA+DMFT framework, Sr2IrO4 can be fully described by a half-filled one-bandHubbard model. The critical value for the Coulomb parameter U is then expected to be between 1.25and 1.5 eV. The metal-insulator transition will appear already at moderate values of the Coulombrepulsion U ∼ 2 eV.

Conclusion of this density functional study

In conclusion, the spin-orbit coupling and the distortions in Sr2IrO4 both induce complementary effectsin the electronic band structure of Sr2IrO4. Whereas the former modifies strongly the distribution ofthe t2g bands so that almost one band remains close to the Fermi level – the “jeff = 1/2 band” – ,the latter induces an hybridization between the dxy and the dx2−y2 band which leads to a separation

11We remind again to the reader the following numerical values:√

1/6 ≈ 0.40824,√

1/3 ≈ 0.57735 and√

2/3 ≈0.81649.


Γ M X Γ -3

-2

-1

0

1

2

Ene

rgy(

eV)

0

0.2

0.4

0.6

0.8

1

Γ M X Γ-3

-2

-1

0

1

2

Ene

rgy

(eV

)

Figure 5.25: Magnitude of the Wannier character |ψ3〉 along the Kohn-Sham band structure of Sr2IrO4

in the energy window [−3; 2] eV. The scale in renormalized arbitrary unit gives the weight of theWannier character in the band. This Wannier character can be associated to the “modified ” jeff = 1/2state. The band structure of Sr2IrO4– with distortions and the spin-orbit coupling – is reminded onthe left picture.

0

0.2

0.4

0.6

0.8

1

Γ M X Γ-3

-2

-1

0

1

2

Ene

rgy

(eV

)

0

0.2

0.4

0.6

0.8

1

Γ M X Γ-3

-2

-1

0

1

2

Ene

rgy

(eV

)

Figure 5.26: Same as the right picture above for the Wannier characters |ψ4〉 (left picture) and |ψ5〉(right picture). While |ψ4〉 can be considered as the “modified ” jeff = 3/2 mj = −1/2 character, thecharacter jeff = 3/2 mj = +3/2 can be associated to |ψ5〉.

5.2. EFFECTS OF THE SPIN-ORBIT COUPLING & THE DISTORTIONS WITHIN LDA+DMFT101

in energy between eg and t2g bands and to an orbital polarization – the bonding “dxy band” becomescompletely filled –.

These two effects alone are not efficient enough to reach a Mott insulating state, when the electroniccorrelations are also taken into account. However, when these two phenomena are applied together –like in the realistic compound Sr2IrO4 –, they enable to isolate one band – or more precisely, four bandsbecause of the foldings – close to the Fermi level whose bandwidth is about 1 eV. A Mott insulatingstate then emerges due to electronic correlations: Sr2IrO4 can thus be understood as a Mott insulator,driven by the joint effort of the spin-orbit interaction and the distortions.

To confirm this picture and quantify more precisely the help given by the spin-orbit coupling andthe distortions to the electronic correlations, an LDA+DMFT calculation was carried out in each ofthe four cases we have previously studied. The results are presented and discussed in the followingsection.

5.2 Influence of the spin-orbit coupling and the distortions on elec-tronic correlations: LDA+DMFT study

In the previous section on LDA calculations, we have already highlighted the effect of the spin-orbitcoupling and the structural distortions on the Kohn-Sham band structure of Sr2IrO4. These effects giveus some ideas on how the spin-orbit coupling and the distortions will make the electronic correlationsmore efficient, but so far we did not quantify this. In this part, we determine the impact of electroniccorrelations by performing LDA+DMFT calculations for all the cases shown in the schematic plot 5.2.Doing the calculations separately for spin-orbit coupling and distortions, similar to the previous sec-tion, we can distinguish clearly between their influence, and in a final step, we analyze the correlationsfor the realistic situation, where both spin-orbit and distortions are present.

When performing DMFT calculations, a problem that arises is the proper determination of theinteraction parameters: the Coulomb intra-orbital repulsion U , the Coulomb inter-orbital repulsion U ′

and Hund’s rule exchange coupling J (cf. equation (1.36)). So far, there is no estimate from theoryfor these parameters – which could in principle be done within constrained-RPA or similar methods –that is why we have chosen to use U and J as parameters, assuming that a lattice model restrictedonly to t2g orbitals still holds for all the considered cases.

In the following, we thus perform LDA+DMFT calculations for several values of the parameterU , ranging from about 1 eV to 6 eV, in order to study the metal-insulator transition. By doing this,we are able to get some information on the critical U , when the transition occurs. This is a way to“quantify” the efficiency of spin-orbit coupling and distortions to drive the system into an insulatingstate. Moreover, we can analyze in details, how the transition occurs. In particular, we are interestedin how the orbital polarization, the degeneracy of the atomic multiplet, or the narrowing of the effectivebandwidth influence the metal-insulator transition, and how these properties change with spin-orbitcoupling and the presence of distortions.

With this idea in mind, our purpose will be to treat all the cases introduced in the previous section(undistorted without spin-orbit coupling, distorted without spin-orbit coupling, undistorted with spin-orbit coupling, distorted with spin-orbit coupling) at roughly the same conditions, and to look for thecritical value of U . Remember that the goal is to find a situation, where a realistic value of U ≈ 2 eVis sufficient to get an insulating state. As for all multiband systems, the Hund’s rule exchange Jmust not be neglected (cf. equation (1.37)). For a 5d transition metal oxide, it is expected that J isnot larger than 0.2 eV. That is why we fix J to this value, keeping only U as a parameter. This is


also justified because J is not affected too much by electronic screening, so the material dependenceof this parameter should be weak. To speed up the calculations, we do calculations using density-density interactions only, and neglect off-diagonal components of the interaction matrix like spin-flipand pair-hopping terms. Moreover, all calculations in this section have been done at room temperatureT=300 K (which corresponds to β = 1/kBT = 40 eV−1).

5.2.1 Case 1: “Undistorted” Sr2IrO4 without spin-orbit coupling

General description of the metal-insulator transition

In part 5.1.1, the LDA study of undistorted Sr2IrO4 has shown that this idealized compound can bedescribed as a simple t2g system. Within this picture, the three t2g bands of the system accommo-date 5 electrons (by formula unit) and the two eg bands are considered as completely empty. Thisapproximation relies on the two following arguments:

• The charge of the two eg bands is negligible (less than 0.02 electrons according to table 5.3) incomparison to the charge of the t2g bands.

• Electronic correlations will push the eg bands up and leave them completely empty. Since theirbottom is close the the Fermi level, even a small Coulomb repulsion parameter U will be sufficient.

In order to justify these assumptions, we also performed calculations for the full five-band model, in-cluding both t2g and eg states, and the validity of above arguments was indeed confirmed.

In the following, we use only the t2g bands to build the local Hamiltonian of our problem. Atechnical reason for this simplification is that reducing the number of orbitals from 5 to 3 makesthe calculations one order of magnitude faster, since the dimensionality of the local problem scalesexponentially with the number of orbitals. In order to carry out the LDA+DMFT calculations, weuse the standard implementation of LDA+DMFT within the FP-LAPW framework of Aichhorn etal. [1], which was described in chapter 2. Moreover, since the spin-orbit coupling is not included, thecorresponding impurity problem can be solved by the CTQMC method without any problems. Asexplained in section 3.4, an analytical continuation is needed in order to obtain results on the real-frequency axis. The details to finally get the lattice spectral function have been presented in section 3.4.

Since only the t2g bands are considered, the Wannier functions were constructed from the energywindow [−3.5; 0.6] eV, as introduced in part 5.1.1. According to their charge given in table 5.3, thetotal charge of the impurity model is 5.004 electrons. Moreover, since there is no hybridization termbetween the Wannier functions, the CTQMC calculation is free of any sign-problem. We carried outLDA+DMFT calculations for this system for Coulomb repulsion U ranging from 2 to 6 eV.

The total spectral density obtained for U=2, 3, 3.8 and 4 eV are shown in figure 5.27. In all the plotsthat will be shown in the remainder of this thesis, the Fermi level is set to 0 eV. Our calculations showthat the metal-insulator transition occurs between 3.8 and 4.0 eV. This value confirms that undistortedSr2IrO4 would be a rather weak correlated metal for a realistic value of U of the order of 2 eV. In thepresent case, a Coulomb parameter of slightly below 4 eV is necessary to reproduce the measured gapof about 0.3 eV.

An insulating state with 2 bands 3/4-filled

To go further in the understanding of this metal-insulator transition, the spectral density obtainedfor the different orbitals are depicted in figures 5.28 and 5.29. Since the orbital dxz and dyz are de-generate, only the results for the dxy and the dxz Wannier functions are plotted. The left panels of

5.2. EFFECTS OF SPIN-ORBIT & DISTORTIONS WITHIN LDA+DMFT 103

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5Energy (eV)

0

2

4

6

8

10

12

14

DO

S (s

tate

s.eV

-1)

LDAU=2 eVU=3 eVU=3.8 eVU=4 eV

Figure 5.27: Total LDA+DMFT spectral functions of undistorted Sr2IrO4 without spin-orbit couplingfor U=2, 3, 3.8 and 4 eV. The LDA DOS is shown as dashed line, and the Fermi level is set to 0 eV.

figures 5.28 and 5.29 show the spectral function in the metallic regime (U≤ 3.8 eV), whereas the rightpanels display the results in the insulating regime. Both orbitals (dxy and dxz) undergo their metal-insulator transition simultaneously – between 3.8 eV and 4.0 eV.

On the left panels, the well-known three-peak structure can be easily recognized: the quasi-particlepeak is located close to the Fermi level, surrounded by the incoherent part of the spectrum whichcorresponds to the two Hubbard bands. As the value of the Coulomb parameter U increases, thequasi-particle peak becomes narrower and the corresponding spectral weight is transferred over largeenergy scales to the Hubbard bands. Moreover, the internal structure of the quasi-particle peak is lesspronounced as the metal-insulator transition approaches: the same structure as the original DOS ofthe Wannier orbital appears to be rescaled for U=2 eV, whereas for U=3 eV, only peaks which remindthe Van-Hove singularities remain, before merging for higher values.

This behavior of the spectral functions are directly related to the properties of the associatedself-energies by the relations established in chapter 2. However, in this present case, they can besimplified, as shown in Appendix B. As a result, the spectral density associated to each t2g characterm (m = dxy, dxz, dyz) can be written as:

Aα,σm (ω) = − 1

π

∑

k∈BZ

Im

[Dα,σm (k)

ω + i0+ + µ− εσkνm

−∆Σα,σm (ω)

](5.7)

where Dα,σm (k) is the LDA partial DOS associated to the spin σ and the m character along the “m

t2g band” and ∆Σα,σm (ω) is the local self-energy associated to the m Wannier orbital. The spectralfunctions can thus be considered as independent one-band spectral function with respect to their cor-responding character and the physical interpretation already presented in chapter 1 for the one-bandmodel can be applied here for each orbital separately.

The imaginary parts of the self-energies are displayed on figures 5.30. From their linear regime atsmall Matsubara frequencies, the quasi-particle weight Z can be estimated, and the results are givenin table 5.7. The width of the quasi-particle peak is then ZW , where W is the initial bandwidth of


-5 -4 -3 -2 -1 0 1 2 3Energy (eV)

0

0.25

0.5

0.75

1

1.25

1.5

DO

S (s

tate

s.eV

-1)

Wannier orbitalU=2 eVU=3 eVU=3.8 eV

-5 -4 -3 -2 -1 0 1 2 3Energy (eV)

0

0.25

0.5

0.75

1

1.25

1.5

DO

S (s

tate

s.eV

-1)

Wannier orbitalU=4 eVU=5 eVU=6 eV

Figure 5.28: LDA+DMFT spectral functions of the dxy Wannier orbital in undistorted Sr2IrO4 withoutspin-orbit coupling for U=2, 3, and 3.8 eV (left panel) and U=4, 5 and 6 eV (right panel). Thecalculations were performed at T=300 K (β = 40 eV−1) and with J=0.2 eV. The LDA DOS is shownas dashed line.

-5 -4 -3 -2 -1 0 1 2 3Energy (eV)

0

0.25

0.5

0.75

1

1.25

1.5

DO

S (s

tate

s.eV

-1)


-5 -4 -3 -2 -1 0 1 2 3Energy (eV)

0

0.25

0.5

0.75

1

1.25

1.5

DO

S (s

tate

s.eV

-1)

Wannier orbitalU=4 eVU=5 eVU=6 eV

Figure 5.29: Same as figure 5.28, but for the dxz orbital.


the Wannier function.12

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6Matsubara frequency iω (eV)

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Im[

Σ(iω

) ] (e

V)

Im[ Σd

xy(iω) ] - U=2 eV

Im[ Σd

xy(iω) ] - U=3 eV

Im[ Σd

xy(iω) ] - U=3.8 eV

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6Matsubara frequency iω (eV)

-1

-0.8

-0.6

-0.4

-0.2

0

Im[

Σ (i

ω) ]

(eV

)

Im[ Σd

xz(iω) ] - U=2 eV

Im[ Σd

xz(iω) ] - U=3 eV

Im[ Σd

xz(iω) ] - U=3.8 eV

Figure 5.30: Imaginary parts of the LDA+DMFT self-energy on the Matsubara axis associated to thedxy (left panel) and the dxz (right panel) Wannier orbitals for U=2, 3, and 3.8 eV. Parameters as infig. 5.28. The dashed lines are the linear extrapolations of the self-energy to iω = 0, their slope α isrelated to the quasi-particle weight Z by the relation Z = (1− α)−1.

Besides, there is no pinning of the spectral function to its non-interacting value at 0 eV. Actually,the value of the spectral function at the new Fermi level depends on the value of the initial DOS atRe[Σ(0)]− µ. In a half-filled one-band model, this quantity is zero because of particle-hole symmetry,hence the pinning at the non-interacting value at 0 eV. In a three-band model with 5 electrons, thissymmetry argument does not hold anymore. Consequently, the different values of Re[Σ(0)] − µ as Uincreases, will induce some shift. It implies a modification of the spectral function at the new Fermilevel. However, the numerical noise close to the Fermi level in figures 5.28 and 5.29 prevent from anythorough analysis.

dxy orbital dxz / dyz orbitalU=2 eV 0.711 0.726U=3 eV 0.496 0.508U=3.8 eV 0.334 0.288

Table 5.7: Quasi-particle weight Z of the dxy anddxz Wannier orbitals. The values were calculatedfrom the linearization of the imaginary part of thecorresponding self-energy close to 0.

dxy orbital dxz / dyz orbitalU=4 eV [-0.28;0.30] eV [-0.26;0.25] eVU=5 eV [-0.63;0.98] eV [-0.67;0.77] eVU=6 eV [-0.99;1.41] eV [-0.98;1.27] eV

Table 5.8: Gap between the upper and the lowerHubbard band of each orbital.

Above the metal-insulator transition only the two Hubbard bands remain, as observed on the rightpanel of figures 5.28 and 5.29. The lower Hubbard band exhibits an internal structure which is certainlya reminiscence of the Van-Hove singularities of the initial DOS. The gap between the two Hubbardbands becomes wider as the Coulomb parameter U is further increased. The boundaries of the gap foreach value of the parameter U are presented in table 5.8.

12For the dxy orbital, the significant energy scale to use as W is not the total bandwidth – 4.1 eV –but rather the partbetween −1 eV and 0.6 eV.


By comparing the right panel of figures 5.28 and 5.29, the spectral weight of the upper Hubbardband of the dxy orbital is smaller than its dxz counterpart, which implies that their respective chargesare different in the insulating state. The charge of each Wannier orbital as function of U is presentedin table 5.9.

Value of U (eV) 0.0 2.0 3.0 3.5 3.7 3.8 4.0 5.0 6.0Charge of the dxy orbital 0.806 0.806 0.810 0.821 0.831 0.853 0.899 0.927 0.950Charge of the dxz / dyz orbital 0.848 0.848 0.847 0.842 0.836 0.825 0.801 0.787 0.776

Table 5.9: Evolution of the charge of each Wannier orbital as function of U . By definition, the valueU = 0 eV corresponds to the charge of the Wannier orbital in LDA.

The displayed values confirm that there is a charge transfer from the dxz and dyz to the dxy or-bital, as the strength of the correlations increases. As a result, in the insulating state of undistortedSr2IrO4 without the spin-orbit coupling, the dxz and dyz orbitals are nearly 3/4-filled, while the dxyband is almost full. The self-energy of a system composed of two 3/4-filled bands in the atomic limitwas calculated in Appendix B: the self-energy associated to the dxz and dyz orbitals was then fittedwith the obtained expression and the quality of the fitting increases as U becomes larger. This allowsto conclude that the insulating state of undistorted Sr2IrO4 without the spin-orbit coupling can bedescribed by this 3/4-filled two-band model as soon as U=4 eV.

As a consequence, undistorted Sr2IrO4 without spin-orbit coupling behaves like an effective two-band system, once the metal-insulator transition has occurred. This result is surprising, since ourLDA calculations have shown that the dxy band is initially less occupied than the dxz and dyz bands,because of the small tetragonal field in the IrO6 octahedra. One could have imagined that electroniccorrelations could increase this crystal field up to a critical value, where the dxz and dyz orbitals arecompletely filled, leaving only one electron in the dxy band. In this hypothetical situation, the criti-cal value for the Coulomb parameter U of this half-filled one-band model lies then between 5.12 and6.15 eV, since the bandwidth of the dxy band is about 4.1 eV.

On the contrary, our numerical simulations show that the metal-insulator transition of the systemoccurs for U between 3.8 and 4 eV. This means that the system thus favors the configuration which“minimizes” the critical value of the Coulomb parameter U , or, in other words, the insulating statewhich involves the least energy given by the electronic correlations. Similar orbital polarization wasfound in lanthanum titanate (LaTiO3) and yttrium titanate (YTiO3) [130].

To sum up, this LDA+DMFT study has shown that electronic correlations induce an orbital po-larization in undistorted Sr2IrO4 so that charge is transferred to the dxy orbital, which gets finallycompletely filled in the insulating state. Please note that in part 5.1.3, we have seen on the level ofthe LDA that the structural distortions in Sr2IrO4 already induce a similar orbital polarization. Thisis already a hint that distortions indeed reduce the critical value of U .

5.2.2 Case 2: Distortions leading to orbital polarization

In part 5.1.3, it was shown that a unit cell which contains four formula-units must be used to describethe structure of distorted Sr2IrO4. However, since the four Ir sites are equivalent – there exists atleast one symmetry operation which transforms one site into another one in the unit cell –, it issufficient to solve the local impurity problem for one of them only. Consequently, the complexity of theimpurity problem is not increased and performing an LDA+DMFT calculation will be only slightly


more time-consuming than previously.

Wannier orbitals with distortions

In the Kohn-Sham band structure of distorted Sr2IrO4, the energy gap between the eg and t2g bandsbetween 0.4 and 1.3 eV suggests to consider only the t2g bands for the description of the system. Theelectronic correlations will indeed merely shift the empty eg bands even more above the Fermi level.The dxy band, however, must be kept in the local problem: although it is almost full, its densityat the Fermi level is actually of the same order of magnitude as those of the dxz and dyz bands, asit can be seen in figure 5.20. Distorted Sr2IrO4 can thus be described as an effective three-band system.

However, there is one important complication as compared to the undistorted case. In part 5.1.3 itwas shown that the band structure of distorted Sr2IrO4 mainly results from the hybridization betweenthe dxy and the dx2−y2 states (a t2g and an eg state). It is thus not possible to consider this compoundas a “pure” t2g-band system and the Wannier functions should be constructed from an atomic basisconsisting of the orbitals dxz, dyzand the bonding linear combination of dx2−y2 and dxy states. Sincewe want to avoid to calculate this bonding orbital, we describe this orbital by projecting to the dxycharacter only. This thus allows to construct the Wannier functions from the t2g atomic basis but it isstill not possible to consider the t2g block as an irreducible representation of the 5d group: the cubicbasis is well-adapted to describe the local problem on Ir site, only if the hybridization term betweenthe dxy and dx2−y2 states can be neglected.

The approach we used to create the Wannier orbitals relies on this last assumption: the energywindow [−2.8; 0.4] eV was considered and the Wannier functions were constructed from the t2g atomicbasis (dxy, dxz and dyz) only. According to their charge displayed in table 5.5, the total charge ofthe considered impurity model is 5.01 electrons and since no hybridization term exist betwen thesethree Wannier functions, the CTQMC calculation is free of any sign-problem during the convergenceprocess13.

The metal-insulator transition in distorted Sr2IrO4

The total spectral density obtained for U=2, 3, 3.2 and 3.4 eV are displayed in figure 5.31. Our cal-culations show that the metal-insulator transition occurs between 3.2 and 3.4 eV. As expected, thisvalue is smaller than the critical value of U in the undistorted case. A Coulomb parameter of 3.4 eVis necessary to get a Mott gap of about 0.3 eV.

In figures 5.32 and 5.33 the orbitally resolved spectral densities are shown (dxz and dyz are degen-erate). The left panels of figures 5.32 and 5.33 depict the spectral functions in the metallic regime(U≤ 3.2 eV), whereas the right panels display the results in the insulating regime. Both orbitals (dxyand dxz) undergo their metal-insulator transition simultaneously between 3.2 eV and 3.4 eV.

As the value of U increases, the evolution of the dxz orbital is qualitatively similar to what we haveobserved in the previous section. The well-known three-peak structure can be easily recognized in theleft panel of figure 5.33, whereas only the two Hubbard bands can be seen in the right panel.

On the contrary, the upper Hubbard band of the dxy orbital is hardly visible, whatever the value forthe Coulomb parameter U is. This results from the filling of the dxy band : Already at the LDA level,the band is almost completely filled, with 0.96 eletrons by unit cell according to table 5.5. Nevertheless,

13Including both dxy and dx2−y2 and their hybridization into the local Hamiltonian was tried but is actually notpossible, because the hybridization introduces a severe sign-problem, making calculations at the temperatures of interestimpossible.


-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5Energy (eV)

0

2

4

6

8

10

12

14

DO

S (s

tate

s.eV

-1)

LDAU=2 eVU=3 eVU=3.2 eVU=3.4 eV

Figure 5.31: Total LDA+DMFT spectral functions of distorted Sr2IrO4 without spin-orbit couplingfor U=2, 3, 3.2 and 3.4 eV. The LDA DOS is plotted as dashed line.

a sharp quasi-particle peak is observed in the metallic regime (U≤ 3.2 eV) and its spectral weight istransferred to the lower Hubbard band as the value of U increases.

As in the undistorted case, the spectral functions of each Wannier orbital can be considered asindependent one-band spectral function with respect to their corresponding character – the proof ofAppendix B still holds. The width of the quasi-particle peak is then ZW , where W is the initial band-width of the Wannier function14 and Z the quasi-particle weight. For U=2 and 3 eV, the values of Z forboth orbitals are given in table 5.10. Contrary to the values displayed in table 5.7, the quasi-particleweight Z of the two orbitals are not equal anymore. Further studies must be carried out to understandwhether these differences can be linked to their different filling.

In the insulating regime (U≥ 3.4 eV), the upper Hubbard band of the dxy orbital can be neglected.Moreover, the upper limits of the lower dxy and lower dxz Hubbard bands are nearly the same. As aresult, the gap of distorted Sr2IrO4 corresponds to the gap between the two dxz Hubbard bands. Itsvalue as function of U are presented in table 5.11.

dxy orbital dxz / dyz orbitalU=2 eV 0.719 0.657U=3 eV 0.474 0.353

Table 5.10: Quasi-particle weight Z of the dxyand dxz orbitals. The values were calculated fromthe linear regime of the imaginary part of the cor-responding self-energy close to 0.

Gap in distorted Sr2IrO4

U=3.4 eV [-0.37;0.15] eVU=4 eV [-0.69;0.12] eVU=5 eV [-1.10;0.58] eV

Table 5.11: Gap between the upper and the lowerHubbard band of the dxz (dyz) orbital, which de-termines the gap in “distorted ” Sr2IrO4

Similar to the previous section, we studied the evolution of the charge of each Wannier orbital asfunction of the value of U . In the metallic regime, they remain constant to their respective initial

14For the dxy orbital, the significant energy scale to use as W is not the total bandwidth – 3.2 eV – but rather thepart between −0.5 eV and 0.25 eV.


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Wannier orbitalU=3.4 eVU=4 eVU=5 eV

Figure 5.32: LDA+DMFT spectral functions of the dxy Wannier orbital in distorted Sr2IrO4 withoutspin-orbit coupling for U=2, 3, and 3.2 eV (left picture) and U=3.4, 4 and 5 eV (right picture).

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Wannier orbitalU=3.4 eVU=4 eVU=5 eV

Figure 5.33: Same as fig. 5.32, but for the dxz orbital.


LDA value, which is 0.968 for the dxy orbital and 0.770 for the dxz (dyz) orbital. Above the the metal-insulator transition the charge of the dxy orbital increases slowly with the value of U , whereas thecharge of the dxz (and dyz) orbital slowly decreases. Therefore, in the insulating regime, the systemcan be described by a dxy band completely filled and two 3/4-filled orbitals (dxz and dyz), this picturebecoming more and more exact as the value of U is further increases.

This result is in good agreement with our expectations: the insulating state in undistorted anddistorted Sr2IrO4 are similar but the metal-insulator transition occurs earlier when the distortions aretaken into account. However, the narrowing of the dxz anddyz bands from 1.45 to 1.40 eV is not enoughto explain this decrease of the critical value from 3.8-4.0 eV to 3.2-3.4 eV. It is the orbital polarizationinduced by the structural distortions which plays the major role. In undistorted Sr2IrO4, the threet2g bands are initially almost equally filled at U = 0 eV. Before being able to open an insulating gap,correlations have to polarize the system first. In distorted Sr2IrO4, since the dxy band is nearly fulleven at U = 0 eV, the electronic correlations need not introduce an additional polarization and areonly used to open the gap, hence a smaller critical value of U . The values of the charge in the table 5.9confirm that between 3.0 and 4.0 eV, the charge transfer is the most important. As a result, the orbitalpolarization can be seen as a “physical requirement ” in the compound to achieve its insulating state.

In addition, in both undistorted and distorted Sr2IrO4, the insulating state with two 3/4-filledbands corresponds to the physical state obtained with the minimal critical value for U . Because ofthe larger bandwidth of the dxy band, opening a Mott gap for the half-filled dxy orbital would requirea larger Coulomb parameter U . Nevertheless, for the sake of completeness, we want to note that thebehavior of the self energy of undistorted and distorted Sr2IrO4 are different in the insulating regime.The imaginary part of the self-energy of the dxz (and dyz) orbital appears to diverge at low frequenciesfor U≥ 3.4 eV in distorted Sr2IrO4. It is thus not possible to fit them with the expression obtainedfor the 3/4-filled two-band model at the atomic limit. This behavior is still unexplained and furtheranalyzes are currently in progress to understand it.

5.2.3 Case 3: Spin-orbit coupling and the reduction of the effective degeneracy

Following part 5.1.2, the local basis at each Ir site in the presence of spin-orbit coupling is given by theeg orbitals (d3z2−r2 and dx2−y2), the jeff = 1/2 and the jeff = 3/2 states. Applying the same argumentsas for Case 1, we neglect the eg sates in the local Hamiltonian and work in the t2g subset only.

Wannier function and local Hamiltonian in the presence of spin-orbit coupling

The method used to numerically find the local basis which describes at best the local problem on anIr site was explained in details in part 5.1.2. It imposes to use a large energy-window – [−3.5; 6.5] eVin this case – and to construct the Wannier functions from the obtained atomic states in this sameenergy-window. The resulting Wannier orbitals were found in one-to-one correspondence with the fiveIr 5d bands. If one wants to use these Wannier projectors directly in the LDA+DMFT approach, onehas to face the following problem. Since we change our basis from cubic harmonics to the jeff = 1/2 andjeff = 3/2 basis, off-diagonal complex hybridizations are introduced to the local Hamiltonian. However,


the current CTQMC impurity solver can not deal with complex numbers yet.15

Nevertheless, in order to perform the LDA+DMFT calculations in the presence of spin-orbit cou-pling, we decided to use a simplified approach. Since there still exits a one-to-one correspondencebetween the bands and the Wannier projectors, the density of the Wannier orbitals corresponds alsoto the DOS of the corresponding band. As a result, it is possible to make the calculation by usingthis DOS in the self-consistency loop. What we are neglecting here are off-diagonal elements of theself-energy, which are of course not captured.

The main difference between this approach and the former implementation we used is that theself-energy is not “upgraded ”. However, since the local self-energy of our local problem is diagonal, thisis not a problem: each band can be seen as treated independently. This method was frequently usedin the past and it was shown that these numerical simulations delay the metal-insulator transition incomparison to the scheme we have up to now used. In other words, it overestimates the critical valueof U [10, 158].

With this other LDA+DMFT implementation, it is now possible to perform the calculation. Asalready explained, the compound can be seen as a one-band model. It is thus tempting to consideronly the jeff = 1/2 Wannier orbital. However, by doing such a calculation, it will not be possible tomake a clear comparison with the three band calculations done in the previous sections. Moreover, thevalue of the DOS at the Fermi level of the jeff = 3/2 bands is of the same order of magnitude as theone of the jeff = 1/2 band. We thus decided to consider the corresponding three band model with thethree Wannier orbitals: jeff = 1/2 and jeff = 3/2.

We remind that there was some problem when calculating the DOS of the Wannier orbital, dueto the integration scheme used in the program. In order to minimize these effects, the DOS wereslightly renormalized: the tail between −4 and −2 eV of the jeff = 1/2 and the jeff = 3/2 |mj | = 1/2orbitals was suppressed, their charge then renormalized: 0.592 instead of 0.573 for the former and0.929 instead of 0.910 for the latter. The charge of the jeff = 3/2 |mj | = 3/2 is still the same (0.913),which leads to a total local charge of 4.87. The total charge of the impurity model was neverthelessset at 5.0 electrons during the calculation. In this case, we chose to perform an analytic continuationof the local Green-function using a stochastic version of the maximum entropy method [19].

The metal-insulator transition in “undistorted” Sr2IrO4 with spin-orbit coupling

The LDA+DMFT spectral densities obtained for each orbital are depicted in figures 5.34 and 5.35 (a)and (b). As expected, the insulating state of the undistorted Sr2IrO4 with spin-orbit coupling corre-sponds to the Mott insulating state of the half-filled jeff = 1/2 Wannier orbital, the two jeff = 3/2Wannier orbitals being completely filled. However, by studying more thoroughly the properties of eachWannier orbital, it appears that three main steps must be considered during the overall metal-insulatortransition in the compound.

15We remind that the Lehmann representation of the Green function Gij(τ) is:

Gij(τ) = −〈ci(τ)c†j(0)〉 =∑

m

〈0|ci|m〉〈m|c†j |0〉 exp(−βEm) (5.8)

If the local Hamiltonian is symmetric and real, its eigenvalues are real and it is possible to find a basis of real eigenvectors.As a result, the Green function Gij(τ) where i and j belong to this real basis is also real according to the previousexpression. If the Hamiltonian is a complex hermitian operator, the eigenvalues are still real, but the eigenvectorswill have complex coefficients. Consequently, the Green function Gij(τ) where i and j belong to an eigenbasis of theHamiltonian is a complex number in general, except of the case i = j, where it is real.


First of all, for the small values of the Coulomb parameter – up to a value U1 between 2 and 3 eV –,the jeff = 3/2 Wannier orbitals are not completely filled yet. As observed on the figures 5.35 (a) and (b),they thus have some weight at the Fermi level. Besides, their respective self-energy on the Matsubaraaxis have a small but non-zero imaginary part. The electronic correlations thus have a weak impacton these states, hence the two features in their spectral functions which remind of a lower Hubbardband and a quasi-particle peak. The jeff = 1/2 Wannier orbital is in a weak correlated metallic stateand as the value of U increases, its charge decreases until one electron remains in the orbital.

For intermediate values – between U1 and the critical value Uc which lies between 3.2 and 3.7 eV –,the final charge repartition is achieved: the two jeff = 3/2 Wannier orbitals are filled and there remainsonly one electron in the jeff = 1/2 Wannier orbital. As observed in the figures 5.35-(a) and (b), thejeff = 3/2 Wannier orbitals are merely shifted above the Fermi level. Their self-energy can be consid-ered as pure real constants and the electronic correlations act only on the jeff = 1/2 Wannier orbitalwhich undergoes its metal-insulator transition at Uc.

For U ≥ Uc, the insulating state is reached. As the Coulomb parameter is further increased, thetwo jeff = 3/2 Wannier orbitals are shifted lower in energy and the gap between the two Hubbardbands of the jeff = 1/2 Wannier orbital becomes larger. The self energy of the jeff = 1/2 Wannierorbital diverges in zero frequency, as expected.

Consequently, the metal-insulator transition of undistorted Sr2IrO4 with the spin-orbit coupling isdirectly the metal-insulator transition of the jeff = 1/2 Wannier orbital which occurs between 3.2 and3.7 eV. However, we would like to point out that the gap corresponds to the gap between the upperjeff = 1/2 Hubbard band and the two filled jeff = 3/2 orbitals, as shown in figure 5.36.

In part 5.1.2 the bandwidth of the jeff = 1/2 orbital was estimated to be of the order of 1.8 eV.Consequently, the critical value for an half-filled one-band model with the same bandwidth is expectedto lie between 2.25 and 2.7 eV. Our numerical simulations seem thus to overestimate the critical valueof U , but note that the calculations were performed on a three-band model. Taking into account thejeff = 3/2 orbitals has delayed the metal-insulator transition, because electronic correlations have alsodriven the filling of these bands in addition of the metal-insulator transition of the jeff = 1/2 orbital.

To emphasize this effect, a half-filled one-band Hubbard model with a DOS of the jeff = 1/2 Wan-nier state was studied as function of the Coulomb parameter U . The LDA+DMFT spectral densitiesobtained are depicted in figure 5.37. As expected, the metal-insulator transition occurs for U = 3 eV,since the bandwidth of the considered Wannier orbital is about 2.0 eV. The delay in the metal-insulatortransition of the three-band model induced by the filling of the jeff = 3/2 bands is then of the order of0.2-0.7 eV.

To sum up, our calculations show that the critical value for the Coulomb parameter U is smallerin undistorted Sr2IrO4 with spin-orbit coupling than in the one without the spin-orbit coupling.16.The spin-orbit coupling thus enhances the effect of electronic correlations by reducing the effectivedegeneracy of the impurity problem. Besides, the metal-insulator transition occurs almost in the sameenergy-range as in distorted Sr2IrO4 without spin-orbit coupling. The jeff = 1/2 Wannier orbital isactually broader by a factor of about

√2 than the dxz and dyz bands. This must compensate for the

increase of the critical value of U when a two-band model is used rather than a one-band model.

16The comparison is not rigorously exact since two different LDA+DMFT methods was used. As explained at thebeginning of this subsection, the critical value we have found are slightly overestimated in comparison to the one wewould have obtained if the same LDA+DMFT implementation as in the case without spin-orbit coupling were used.Fortunately, this precision does not change the conclusion.


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Wannier orbitalU=2 eVU=3 eVU=3.2 eVU=3.7 eV Value of U (eV) 2.0 3.0 3.2

Quasi-particle weight Z 0.552 0.227 0.204

Value of U (eV) 3.7 4.0Gap width (eV) [-0.20;0.15] [-0.29;0.27]

Figure 5.34: LDA+DMFT spectral functions of the jeff = 1/2 Wannier orbital in undistorted Sr2IrO4

with spin-orbit coupling for U=2, 3, 3.2 and 3.7 eV. The quasi-particle weight and the gap of thejeff = 1/2 Wannier orbital are displayed on the right hand side.

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Figure 5.35: Same as fig. 5.34, but for the jeff = 3/2 Wannier orbitals.


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Figure 5.36: LDA+DMFT spectral functions of each Wannier orbital in the undistorted Sr2IrO4 withthe spin-orbit coupling for U=4 eV. The gap lies between the filled jeff = 3/2 orbitals and the upperjeff = 1/2 Hubbard band.

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Figure 5.37: LDA+DMFT spectral density of a half-filled one-band Hubbard model whose DOS is theone of the jeff = 1/2 Wannier state. The calculations were performed for U=2.8, 2.9 and 3.0 eV.


5.2.4 The Mott insulating state in Sr2IrO4

According to its Kohn-Sham band structure (cf. figure 5.1), Sr2IrO4 with both the structural distortionsand the spin-orbit coupling, has only four partially filled bands crossing the Fermi level. As a result, inan LDA+DMFT approach, these four bands will be strongly modified by the electronic correlations,whereas all the other bands will be merely shifted, further above or below the Fermi level. In whatfollows, we restrict ourselves to these four bands and perform an LDA+DMFT calculation only onthem.

Introduction to the model used

In part 5.1.4, it was shown that the four bands which cross the Fermi level in Sr2IrO4 can be understoodas the foldings of one half-filled band only, whose character is a linear combination of the “modified ”jeff = 1/2 and the jeff = 3/2 |mj | = 3/2 states. However, we were not able to find the expression ofthis atomic orbital and the Wannier function which is associated to these four bands is consequentlynot known.

As a result, it is not possible to perform an LDA+DMFT calculation for Sr2IrO4 in the same wayas in subsections 5.2.1 and 5.2.2. In addition, even if this Wannier function was known, the implemen-tation of LDA+DMFT based on the Wannier projectors and the CTQMC impurity solver could nothave been used for the same reasons as part 5.2.3.

However, as it can be seen on the kohn-Sham band structure, these four bands are well-separatedfrom the other bands at each k-point and it is then possible to find an energy window around the Fermilevel which contains only them. As a result, the total DOS in this energy range corresponds exactlyto the DOS associated to these bands. By using this part of the total DOS, we can thus performan LDA+DMFT calculation with the same method as described in part 5.2.3. In other words, sincethe four bands are the replicas of the same band, the problem is finally to solve a half-filled one-bandHubbard model with a particular DOS.

The energy window we have chosen is displayed on figure 5.38. The upper boundary was set to0.56 eV, as given by the Kohn-Sham band structure. The lower boundary was set to −0.35 eV suchthat the obtained DOS is half-filled. As can be seen in the left panel of figure 5.38, with this choice, thefour bands are completely included in the energy window all along the k-path, except for the segment[Γ M ]. Moreover, the upper part of two other bands are slightly included in this energy-range, mostlyin the segment [Γ M ]. The contribution of these two extra bands between −0.25 and −0.35 eV isassumed to be small in the obtained DOS – hatched in purple on the right panel of figure 5.38.

The metal-insulator transition in Sr2IrO4

The LDA+DMFT spectral density obtained for the half-filled one-band model used to describe Sr2IrO4

are displayed in figure 5.39. Since the bandwidth of the initial DOS is about 0.91 eV, a critical valueof the Coulomb parameter between 1.14 and 1.36 eV was expected. Our calculations show that themetal-insulator transition indeed occurs between 1.1 and 1.3 eV.

The evolution of the self-energy and the spectral density as U increases follows the general patterndescribed in section 1.4 with the DOS of a Bethe lattice. The only difference comes from the innerstructure of the quasi-particle peak for U=0.75 eV and of the Hubbard bands for U=1.3 and 1.5 eV.One can see some features of the initial DOS, especially the two peaks at −1 and 0.25 eV. The Mottgap is about 0.2 eV (0.4 and 1.5 eV) for U=1.3 eV (1.5 and 2 eV respectively).


Z Γ M X Γ P N -3

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Figure 5.38: Kohn-Sham band structure (on the left) and total LDA DOS (on the right) of Sr2IrO4.The dashed red lines – at −0.35 and 0.56 eV– define the energy window used to describe Sr2IrO4inan LDA+DMFT approach. It contains essentially the four bands which cross the Fermi level. Thecorresponding part of the total DOS (hatched in purple) is exactly half-filled.

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Figure 5.39: LDA+DMFT spectral density of the half-filled one-band Hubbard model used to describeSr2IrO4 – with distortions and spin-orbit coupling. The calculations were performed for U=0.75, 1.0and 1.1 eV (left panel) and U=1.3, 1.5 and 2.0 eV (right panel). The truncation of the total LDA DOSused has been plotted in dashed line.


As explained previously, all the other bands are either completely empty or filled. It can thus beexpected that in a more precise calculation that includes more bands, the same quantitative resultswill be obtained. With this last argument, it is then possible to draw a comparison between theseresults and those obtained in the previous sections. Although the impact of the spin-orbit couplingand of the structural distortions taken separately on the critical value Uc is weak, the metal-insulatortransition occurs dramatically earlier when both of these two effects are considered.

By adding the spin-orbit corrections to distorted Sr2IrO4, the 3/4-filled two-band impurity problemis transformed in a half-filled one-band problem. This reduction of the effective degeneracy induces afirst decrease in the value of Uc, but the reduction of the bandwidth considered in the model – from1.4 eV to 0.91 eV – plays also a key role in this decrease.

Similarly, by adding the structural distortions to undistorted Sr2IrO4 already described with thespin-orbit coupling, a significant reduction of the bandwidth can be noticed: that is why the metal-insulator transition occurs around 3 eV on figure 5.37 and about three times earlier on figure 5.39.Moreover, the orbital polarization induced by the distortions avoid the delay in the metal-insulatortransition due to the filling of the other bands, which exists in the undistorted case.

To conclude, a Coulomb parameter between 1.3 and 1.5 eV is thus sufficient to get a Mott gap ofabout 0.3 eV, as experimentally found in [115]. Such a value for the Coulomb parameter is physicallypossible in a 5d transition metal oxide, like Sr2IrO4. Therefore, this LDA+DMFT study confirms thatSr2IrO4 is a Mott insulator at room temperature, thanks to the joint effort of the spin-orbit couplingand the structural distortions in the compound.

Conclusion

In this thesis, we have studied the transition metal oxide Sr2IrO4. Although being a 5d material,it exhibits insulating behavior, which is unexpected for an iridium-based compound. In particular,we have studied how structural distortions and spin-orbit coupling modify the electronic structure ofthis compound, and how they make it possible to open an insulating gap for moderate values of theinteraction. According to our calculations, a Coulomb parameter between 1.3 and 1.5 eV is enough toproduce a Mott gap of about 0.3 eV.

These works have also highlighted the importance of taking into account both the spin-orbit couplingand the structural distortions. Moreover, the underlying effects induced by these two parameters,separately and also together, were understood and quantified:

• The structural distortions (the rotations of the IrO6 octahedra of about 11 around the c-axis)introduce an hybridization between the dx2−y2 and the dxy bands, which leads to the formationof a filled bonding and an empty antibonding band and an energy separation between the egand t2g bands. The “distorted ” compound can then be described as a system with two 3/4-filledbands, instead of a complete t2g system. This orbital polarization induced by the distortionsmakes the electronic correlations slightly more efficient than in the corresponding “undistorted ”compound.

• The spin-orbit coupling (of about 0.4 eV on each Ir site) causes a deep change in the “nature”of the t2g bands. They must indeed be seen as the result of the hybridization between the O-2p states and the orbitals of Ir-5d in the limit of strong spin-orbit coupling. As a result, one“jeff = 1/2 band” lies almost alone close to the Fermi level, whereas the two “jeff = 3/2 bands”are almost completely filled. This reduction of the effective degeneracy in the t2g bands couldhave made the electronic correlations more efficient, if the bandwidth of the “jeff = 1/2 band”was not bigger than those of the previous dxz and dyz bands.

• By introducing both the structural distortions and the spin-orbit corrections in the descriptionof Sr2IrO4, one half-filled band – four times folded – remains well separated from all the otherbands. Its bandwidth is narrow enough to get an insulating state with a Coulomb parameterphysically possible for Sr2IrO4. Therefore, the orbital polarization (from the distortions) and thereduction of the effective degeneracy (from the spin-orbit coupling) are not the underlying effectswhich occur but there is also a reduction of the bandwidth which comes from the joint-effort ofthese two parameters. These three effects explain the insulating state in Sr2IrO4.

Furthermore, contrary to what can be found in the litterature, our work has revealed that the“jeff = 1/2 bands” and “jeff = 3/2 bands” are not well separated in the LDA band structure of Sr2IrO4.The structural distortions indeed induce a mixing of these two characters in the four half-filled bandswhich will form the upper and lower Hubbard bands in the material. This finding was recently con-firmed by Watanabe et al. [164] who have calculated the one-particle spectrum of Sr2IrO4 by usinga variational cluster approximation (VCA) and explored the internal electronic structure in the insu-lating state. This new understanding of the internal electronic structure of Sr2IrO4 could explain the

119


decrease of the optical gap with increasing temperature, which was recently observed in Sr2IrO4 byMoon et al. [115]. In this sense, our LDA+DMFT calculations can be considered as the first step of amore general LDA+DMFT study of this material within temperature.

In order to describe Sr2IrO4 within LDA+DMFT, it was necessary to take into account the spin-orbit corrections both in the LDA band structure and in the definition of the correlated orbitals. Con-sequently, we have extended the implementation of LDA+DMFT developed in the LAPW frameworkby Aichhorn et al. [1] such that the Wannier orbitals, which are used to define the impurity problem,may be constructed when the spin-orbit interaction is included. The interest for such an approach goesbeyond the present case of Sr2IrO4 since it could be applied in the future to take into account electroniccorrelations in the description of other 5d-transition metal oxides or even topological band insulators.In this sense, some further developments of this “LDA+SO+DMFT implementation” are already underway. In particular, the CTQMC impurity solver must be extended to deal with a complex hermitianHamiltonian, a necessary improvement to be able to perform a complete LDA+DMFT treatment of acompound including the spin-orbit interaction.

As mentioned above, our work has emphasized the key-role played by the spin-orbit coupling andthe structural distortions to reach the Mott insulating state in Sr2IrO4. However, looking at therecent study on Sr2RhO4, it appears that the spin-orbit interaction can be strongly modified – almostdoubled – by electronic correlations [107]. In this case, the question arises to what extent the Coulombparameters U and J themselves depend on the strength of the spin-orbit interaction. The interplaybetween the spin-orbit interaction and the electronic correlations can thus be much more complex andrequire the development of new models to be investigated in the future.

Part III

Appendices

121

Appendix A

Atomic d orbitals and spin-orbit coupling:Complements

In Sr2IrO4 and many other transition metal oxides1 the transition metal ion M is surrounded by sixoxygen atoms O, forming then an MO6 octahedron. As a result, the system is locally of cubic – oreven lower – symmetry and the dx2−y2 , d3z2−r2 , dxy, dxz and dyz orbitals are usually introduced todescribe the atomic states. Moreover, because of this octahedral environment, the fine structure ofthe d orbitals is not the same as in the case of a single atom. This was particularly highlighted insection 3.1 where the jeff = 1/2 and jeff = 3/2 states were introduced.

In this appendix, we first briefly remind the definitions of the dx2−y2 , d3z2−r2 , dxy, dxz and dyzorbitals and then investigate further the fine structure of the d orbitals in two cases:

• in an octahedron with perfect cubic symmetry: In this case, we calculate the atomicd states without any TP-equivalence approximation, in order to emphasize the impact of thenon-diagonal terms neglected in section 3.1.

• in an elongated or compressed octahedron: Within the TP-equivalence approximation, weshow the influence of the additional tetragonal splitting Q1 on the definition of the jeff = 1/2and jeff = 3/2 states.

The results of this latter study are finally used to estimate the value of the spin-orbit coupling constantζSO and the energy splitting Q1 in “undistorted ” Sr2IrO4.

A.1 Atomic d states of a metal in an octahedral ligand field

We consider an atom M surrounded by six point charges −Ze with Z < 0 (for oxygen atoms, Z =−2). The M atom is at the center of the coordinate system, whereas the point charges have thefollowing positions (a1, 0, 0), (−a4, 0, 0), (0, a2, 0), (0. − a5, 0), (0, 0, a3) and (0, 0,−a6), forming thenan octahedron.

Case of an octahedron with “perfect” cubic symmetry

If all the point charges are at the same distance a of the center ion M (∀ i ai = a), the symmetryof the system is “cubic”. The degeneracy of the five d atomic orbitals of the metal is lifted and twomultiplets are created:

1For instance, the transition metal oxides which crystallize in the perovskyte or K2NiF4-type structure

123

124 APPENDIX A. ATOMIC D ORBITALS AND SPIN-ORBIT COUPLING: COMPLEMENTS

• the three t2g states dxy,dxz and dyz:

ϕndxy(r) = − i√2

[Y 22 (r)− Y 2

−2(r)]Rnd(r) =

(15

4π

) 12 (xy

r2

)Rnd(r)

ϕndyz(r) =i√2

[Y 21 (r) + Y 2

−1(r)]Rnd(r) =

(15

4π

) 12 (yz

r2

)Rnd(r)

ϕndxz(r) = − 1√2

[Y 21 (r)− Y 2

−1(r)]Rnd(r) =

(15

4π

) 12 (xz

r2

)Rnd(r)

(A.1)

• the two eg states d3z2−r2 and dx2−y2 :

ϕnd3z2−r2(r) = Y 2

0 (r) Rnd(r) =

(15

16π

) 12(3z2 − r2

r2

)Rnd(r)

ϕndx2−y2(r) =1√2

[Y 22 (r) + Y 2

−2(r)]Rnd(r) =

(15

16π

) 12(x2 − y2

r2

)Rnd(r)

(A.2)

where r denotes the length of the vector r and the angles, θ and φ, specifying the direction of the vectorin spherical coordinates, are indicated as r. n is the principal quantum number (typically n = 3, 4 or 5).

As observed on figure A.1, the t2g orbitals do not point towards the point charges – or the ligandatoms –, in contrast to the eg ones. As a result, the latter are destabilized by an higher Coulombrepulsion energy. More precisely, it can be shown [157] that:

εt2g = ε0 +6Ze2

a− 4Dq and εeg = ε0 +

6Ze2

a+ 6Dq (A.3)

where ε0 is the energy of the d states of the original M atom and the parameters D, q are defined asfollows:

D =35Ze2

4a5and q =

2

105

∫r4|Rnd(r)|2 r2dr (A.4)

A.1.1 Case of an elongated or compressed octahedron

As mentioned in section 5.1, the IrO6 octahedra are slightly elongated along the z-axis in Sr2IrO4.Consequently, the site symmetry of the metal ion is lower than cubic and the degeneracies betweenthe two eg and the three t2g states are lifted.

More generally, we will consider here the case of an elongation – or a compression – of the octahe-dron along the z-axis. In other words, the MO6 octahedron is now defined by a1 = a2 = a4 = a5 = aand a3 = a6 = b, with b 6= a. The symmetry of the system is now “tetragonal ”: the d3z2−r2 , dx2−y2 ,dxy, dxz and dyzorbitals are still eigenstates of the problem but some additional energy splittings occurbetween them, as depicted on figure A.2.

When b < a, the MO6 octahedron is compressed. As a result, the dxz, dyz and d3z2−r2 orbitals“feel” a slightly higher Coulomb repulsion energy than their dxy and dx2−y2 counterparts. Conversely,when b > a, the octahedron is elongated and the latter are now higher in energy than the former. Theenergy splitting induced by the tetragonal field between the two eg (three t2g) states is noted Q2 (Q1

respectively). Their expression in function of the ligand-field can be found in [157]. Moreover, sincethe symmetry of the system is now “tetragonal ”, the d atomic orbitals can not be referred to as the egand t2g states anymore:

A.1. ATOMIC D STATES OF A METAL IN AN OCTAHEDRAL LIGAND FIELD 125

Figure A.1: Angular behaviors of the d atomic orbitals dxz, dyz, dxy (first line) and dx2−y2 , d3z2−r2(second line).

Figure A.2: Energy splitting between the d atomic orbitals in a compressed (panel a), a “perfect” (panelb) and an elongated (panel c) octahedron.


• d3z2−r2 must be called the a1g state and dx2−y2 the b1g state,

• dxz and dyz are now the eg states and dxy the b2g state.

As a result, the Ir-5d dxy orbital must be higher in energy – and thus slightly less populated – thanits dxz and dyz counterparts in Sr2IrO4. This effect is actually observed in table 5.3, when neither thespin-orbit interaction nor the distortions are taken into account.

A.2 Cubic symmetry and spin-orbit coupling

In section 3.1, we have calculated the impact of the spin-orbit interaction on the d atomic states in cubicsymmetry within the TP-equivalence approximation. For the sake of completeness, we present here thecomplete calculation of the fine structure of the d atomic states in cubic symmetry. In addition, westudy the influence of a tetragonal energy splitting Q1 on the definition of the jeff = 1/2 and jeff = 3/2states.

A.2.1 Effect of the spin-orbit interaction beyond the TP-equivalence approxima-

tion

Using the expression of the orbital angular momentum l in the basis dxz, dyz, dxy, d3z2−r2 , dx2−y2– presented in (3.20) –, the complete matrix of the spin-orbit interaction can be reduced to two five-dimensional submatrices:

0 −i i√3 −1

i 0 −1 −i√3 −i

−i −1 0 0 −2i√3 i

√3 0 0 0

−1 i 2i 0 0

.ζSO2

and

0 i i −√3 1

−i 0 1 −i√3 −i

−i 1 0 0 2i

−√3 i

√3 0 0 0

1 i −2i 0 0

.ζSO2

(A.5)

in the basis:

dxz ↑, dyz ↑, dxy ↓, d3z2−r2 ↓, dx2−y2 ↓ and dxz ↓, dyz ↓, dxy ↑, d3z2−r2 ↑, dx2−y2 ↑ respectively.

When we used the “TP-equivalence” approximation in section 3.1, the cubic crystal field ∆ = 10Dqwas assumed much larger than the spin-orbit coupling constant ζSO. As a result, the eg and t2g orbitalswere considered as decoupled and:

• the eg states were not affected by the spin-orbit coupling (εeg = εt2g +∆)

• the t2g states were split into a doublet of eigenstates of energy: εjeff= 12= εt2g + ζSO

|12,−1

2〉 = |jeff =

1

2,mj = −1

2〉 =

1√3|dyz ↑〉 −

i√3|dxz ↑〉 −

1√3|dxy ↓〉

|12,+

1

2〉 = |jeff =

1

2,mj = +

1

2〉 =

1√3|dyz ↓〉+

i√3|dxz ↓〉+

1√3|dxy ↑〉

(3.27)

and a quartet of eigenstates of energy: εjeff= 32= εt2g − ζSO/2

|32,−3

2〉 = |jeff =

3

2,mj = −3

2〉 =

1√2|dyz ↓〉 −

i√2|dxz ↓〉

|32,+

3

2〉 = |jeff =

3

2,mj = +

3

2〉 = − 1√

2|dyz ↑〉 −

i√2|dxz ↑〉

|32,−1

2〉 = |jeff =

3

2,mj = −1

2〉 =

1√6|dyz ↑〉 −

i√6|dxz ↑〉+

√2

3|dxy ↓〉

|32,+

1

2〉 = |jeff =

3

2,mj = +

1

2〉 = − 1√

6|dyz ↓〉 −

i√6|dxz ↓〉+

√2

3|dxy ↑〉

(3.28)

A.2. CUBIC SYMMETRY AND SPIN-ORBIT COUPLING 127

In order to calculate the complete fine structure of the d atomic states in cubic symmetry, wewill use these states to partially diagonalize the complete Hamiltonian H of the system. Indeed, withξ = 2∆/ζSO, one can write:

H = εt2g .Id+

2 0 0 0 0

0 −1 i√6 0 0

0 −i√6 ξ 0 0 O

0 0 0 −1 −i√6

0 0 0 i√6 ξ

2 0 0 0 0

0 −1 −i√6 0 0

O 0 i√6 ξ 0 0

0 0 0 −1 i√6

0 0 0 −i√6 ξ

.ζSO2

(A.6)

in the basis

|12,−1

2〉, |3

2,+

3

2〉, |d3z2−r2 ↓〉, |3

2,−1

2〉, |dx2−y2 ↓〉, |1

2,+

1

2〉, |3

2,−3

2〉, |d3z2−r2 ↑〉, |3

2,+

1

2〉, |dx2−y2 ↑〉

Finally, the fine structure of the d atomic states in cubic symmetry is then composed of:

• a first quartet of eigenstates associated to the energy

ε+ = εt2g +ζSO2

(ξ − 1

2+√6

√1 +

(1 + ξ)2

24

)(A.7)

x |32,−1

2〉+ y |dx2−y2 ↓〉

x |32,+

3

2〉 − y |d3z2−r2 ↓〉

and

x |32,−3

2〉+ y |d3z2−r2 ↑〉

x |32,+

1

2〉 − y |dx2−y2 ↑〉

(A.8)

• a doublet of eigenstates of energy εjeff= 12= εt2g + ζSO

|12,−1

2〉 and |1

2,+

1

2〉 (A.9)

• a second quartet of eigenstates associated to the energy

ε− = εt2g +ζSO2

(ξ − 1

2−√6

√1 +

(1 + ξ)2

24

)(A.10)

−y |32,−1

2〉+ x |dx2−y2 ↓〉

−y |32,−3

2〉+ x |d3z2−r2 ↑〉

and

y |32,+

1

2〉+ x |dx2−y2 ↑〉

y |32,+

3

2〉+ x |d3z2−r2 ↓〉

(A.11)

In the previous expressions, we have used the following quantities :

ξ =2∆

ζSO, h =

ξ + 1

2√6, x =

1√2

√1− h√

1 + h2and y =

i√2

√1 +

h√1 + h2

(A.12)

If ∆ > 0, it can be shown that ε+ > εj= 12> ε−, and when ξ → +∞, the multiplets obtained within

the TP-equivalence approximation are of course found.

As a result, no other splitting is introduced by going beyond the TP-equivalence approximation:the d orbitals are still split into two quartets and one doublet. Moreover, the jeff = 1/2 states are stilleigenstates of the complete Hamiltonian H, whereas the jeff = 3/2 and eg states are mixed.


A.2.2 Effect of a tetragonal splitting within the TP-equivalence approximation

We consider now that the six ligands surrounding the metal ion form a slightly elongated – or com-pressed – octahedron. Moreover, we assume that the cubic crystal field is still much larger than thespin-orbit couping constant (∆ = 10Dq ≫ ζSO).

As previously explained, the d3z2−r2 , dx2−y2 , dxy, dxz and dyz orbitals are still suitable to describethe atomic d states in tetragonal symmetry. The spin-orbit interaction matrix is thus the same as inthe cubic case and the “TP-equivalence” approximation still holds for this system: the d3z2−r2 anddx2−y2 states can be studied independently of the dxy, dxz and dyz states.

As in the cubic symmetry, the spin-orbit interaction is ineffective on the d3z2−r2 and dx2−y2 orbitalsbecause of their quenched angular momentum. On the contrary, the tetragonal splitting Q1 – betweenthe dxy and dxz,dyz states – modifies the structure of the multiplets jeff = 1/2 and jeff = 3/2. In the“t2g” subspace, the Hamiltonian H of the system can indeed be written as follows:

H = εt2g .Id+

0 −i ii 0 −1 O

−i −1 η

0 −i iO i 0 −1

−i −1 η

.ζSO2

in the basis dxz ↑, dyz ↑, dxy ↓, dxz ↓, dyz ↓, dxy ↑ with η = 2Q1/ζSO.Using the same approach as in section 3.1 to diagonalize this matrix, one obtains three doublets of

eigenstates:

• a first doublet of eigenstates associated to the energy

εjeff= 12= εt2g +

ζSO2

(1 + η

2+√2

√1 +

(η − 1)2

8

)(A.13)

|jeff =1

2,mj = −1

2〉 = x√

2|dyz ↑〉 −

ix√2|dxz ↑〉+ y |dxy ↓〉

|jeff =1

2,mj = +

1

2〉 = x√

2|dyz ↓〉+

ix√2|dxz ↓〉 − y|dxy ↑〉

(A.14)

• a second doublet of eigenstates associated to the energy

εjeff= 32,|mj |=

32= εt2g −

ζSO2

(A.15)

|jeff =3

2,mj = −3

2〉 = 1√

2|dyz ↓〉 −

i√2|dxz ↓〉

|jeff =3

2,mj = +

3

2〉 = − 1√

2|dyz ↑〉 −

i√2|dxz ↑〉

(A.16)

• a third doublet with energy

εjeff= 32,|mj |=

12= εt2g +

ζSO2

(1 + η

2−√2

√1 +

(η − 1)2

8

)(A.17)

|jeff =3

2,mj = −1

2〉 = − y√

2|dyz ↑〉+

iy√2|dxz ↑〉+ x |dxy ↓〉

|jeff =3

2,mj = +

1

2〉 = y√

2|dyz ↓〉+

iy√2|dxz ↓〉+ x |dxy ↑〉

(A.18)

A.3. EVALUATION OF THE SPIN-ORBIT COUPLING CONSTANT AND THE TETRAGONAL SPLITTING IN

In these expressions, we have used the following quantities :

η =2Q1

ζSO, h =

η − 1

2√2, x =

1√2

√1− h√

1 + h2and y = − 1√

2

√1 +

h√1 + h2

(A.19)

The states were labeled with the effective quantum number jeff, mj which can be associated to themin the “perfect” cubic symmetry (Q1 = 0).

-10 -8 -6 -4 -2 0 2 4 6 8 10 η=2Q

1/ ζ

SO

-10

-8

-6

-4

-2

0

2

4

6

8

10

2(ε

−ε0)/ζ

SO

j=1/2 - |mj|=1/2

j=3/2 - |mj|=3/2

j=3/2 - |mj|=1/2

Figure A.3: Evolution of the eigenvalues of the system with the amplitude of the tetragonal splittingQ1. The eigenvalues are given with respect to the local energy of the t2g states ε0. All the values areadimensionalized with respect to the value ζSO/2.

The evolution of the energy of each doublet with respect to the parameter η is presented onfigure A.3. Although the jeff = 1/2 states are always the highest in energy, the order of the twojeff = 3/2 doublets depends on the sign of the tetragonal splitting Q1:

• for an elongated octahedron (Q1 > 0), the jeff = 3/2 |mj | = 1/2 states are higher in energy thanthe jeff = 3/2 |mj | = 3/2;

• for a compressed octahedron (Q1 < 0), the jeff = 3/2 |mj | = 3/2 states are higher in energy thanthe jeff = 3/2 |mj | = 1/2;

• for Q1 = 0, they form a degenerate quartet of eigenstates, since the symmetry is merely cubic.

A.3 Evaluation of the spin-orbit coupling constant and the tetragonalsplitting in “undistorted” Sr2IrO4

As mentioned in section 5.1.2, we have estimated the value of the spin-orbit coupling constant ζSO andthe tetragonal splitting Q1 in “undistorted ” Sr2IrO4. To perform these calculations, we have used thepreviously obtained expressions of the jeff = 1/2 and jeff = 3/2 states when the tetragonal splittingQ1 is taken into account.


Estimation of the spin-orbit coupling constant ζSO

Let λ be the energy splitting between the jeff = 1/2 and jeff = 3/2 |mj | = 3/2 bands:

λ = εjeff= 12− εjeff= 3

2,|mj |=

32

(A.20)

From the expressions (A.13) and (A.15), one obtains:

2λ

ζSO=

1 + η

2+√2

√1 +

(η − 1)2

8+ 1 ⇔ η =

Q1(3λ−Q1)

λ(λ−Q1)(A.21)

As a result, if Q1 is known, it is possible to evaluate the value of ζSO, since η = 2Q1/ζSO.

We first applied the formula (A.21) by using the values taken by the bands at the Γ point in undis-torted Sr2IrO4. From figure 5.8, we measured Q1 = −2.45 eV and from figure 5.11, λ = 0.26 eV. As aresult, we found ζSO = 0.436 eV.

We also performed the same calculation by using the mean-energy associated to the correspondingbands. The mean-energy was obtained from:

E =

∫εD(ε)dε∫D(ε)dε

(A.22)

where the integration of the partial DOS was performed between −3.5 and 6.5 eV.2 From the partialDOS of dxy, dxz and dyzdisplayed in figure 5.9, we estimated Q1 = −0.216 eV. From the partial DOSassociated to the jeff = 1/2 and jeff = 3/2 mj = 3/2 states – depicted in figure 5.15– , we obtainedλ = 0.476 eV. This led to ζSO = 0.401 eV. Both these results are in good agreement with the standardestimation of ζSO for iridium [53, 43, 161].

Evaluation of the tetragonal splitting Q1

The previous estimations of Q1 merely are effective tetragonal splittings between the dxy and thedxz,dyz bands. It is however possible to find the “real” value of Q1 by using the expressions of thejeff = 1/2 and jeff = 3/2 |mj | = 1/2 states numerically found in section 5.1.2:

|φjeff= 12,mj=− 1

2〉 = −i0.00588 |φdx2−y2↓

〉 −0.62942 |φdxy↓〉 −i0.54945 |φdxz↑〉 +0.54945 |φdyz↑〉|φjeff= 3

2,mj=− 1

2〉 = −i0.13967 |φdx2−y2↓

〉 +0.76996 |φdxy↓〉 −i0.44026 |φdxz↑〉 +0.44026 |φdyz↑〉(A.23)

By neglecting the term in |φdx2−y2↓〉, we can map this to the expressions (A.14) and (A.18) and then

determine the value of the ratio η. The following relation can indeed be found from (A.19),

η =√2

(x

y− y

x

)+ 1 (A.24)

By applying this formula for the jeff = 1/2 states, we found η = 0.400 and for the jeff = 3/2|mj | = 1/2 states, we obtained η = 0.395. Using then the value of the spin-orbit coupling constantfound previously, we get that Q1 = 0.08 eV.

This value has the same order of magnitude than the one used by Jin et al. [79] to fit the LDAband structure with a tight-binding model. However, their estimation (Q1 = 0.15 eV) is twice biggerthan ours, certainly because we do not use the same crystallographic data.

2This is the energy window in which the Wannier orbitals were calculated.

Appendix B

The self-energy of Sr2IrO4

In this appendix, some properties of the self-energies of Sr2IrO4 are presented. Our study is restrictedto the easier case where both the spin-orbit coupling and the structural distortion are not takeninto account. First, we establish the relation (5.7) which relates the momentum-resolved spectralfunction Aα,σm (k, ω) and the local self-energy ∆Σα,σm (ω) in the particular case when there is a one-to-one correspondence between the Wannier characters (α, lm, σ) and the bands. We then calculate theself-energy of a 3/4-filled two-bands Hubbard model in the atomic limit and show the good agreementbetween the self-energies of undistorted Sr2IrO4 without the spin-orbit coupling with the obtainedexpressions when the Coulomb parameter U is large enough.

B.1 Relation between the spectral function and the self-energy

In a general LDA+DMFT scheme1, the spectral functions Aα,σm,m′(k, ω) is linked to the behavior of theself-energies of the impurity model ∆Σα,σm (ω) by the following relations introduced in section 3.4:

Aα,σm,m′(k, ω) = − 1

πIm

∑

ν,ν′

3∑

j=1

Θα,σlmνj(k)G

σνν′(k, ω + i0+)

[Θα,σlm′ν′j(k)

]∗

= − 1

π

∑

ν,ν′

3∑

j=1

Im

Θα,σlmνj(k)

[Θα,σlm′ν′j(k)

]∗

(ω + i0+ + µ− εσkν)δνν′ − Σσνν′(k, ω)

(B.1)

where Σσνν′(k, ω) =∑

α,mm′

[Pα,σlm,ν(k)

]∗[∆Σαimp(ω)]

σmm′ P

α,σlm′,ν′(k)

and [∆Σαimp(ω)]σmm′ = [Σα

imp(ω)]σmm′ − [Σα

dc(ω)]σmm′ .

In the case of undistorted Sr2IrO4 without the spin-orbit coupling, we have highlighted in part 5.1.1that there is a one-to-one correspondence between the Wannier functions and the t2g bands. In addition,the impurity Green function Gα

loc(iωn) of the t2g block has the same structure as the density matrix : itis diagonal and degenerate in spin (cf. part 5.2.1). As a result, so are the impurity and double-countingself-energies (Σα

imp(iωn) and Σαdc(iωn)). Consequently, if νp refers to the band associated to the pth

Wannier orbital, we can write2 that:

Σσνν′(k, ω) =

[[Σα,σimp

]pp(ω)−

[Σα,σdc

]pp(ω)

]δ(ν − νp)δνν′ = ∆Σα,σp (ω)δ(ν − νp)δνν′ (B.2)

1We do not take into account the spin-orbit coupling in the following study.2We remind to the reader that there is only one Ir atom in the unit cell, so α = 1 in (B.1).

131

132 APPENDIX B. THE SELF-ENERGY OF SR2IRO4

This implies that Aα,σm,m′(k, ω) = Aα,σm (k, ω)δmm′ and:

Aα,σm (k, ω) = − 1

π

∑

ν,ν′

3∑

j=1

Im

[Θα,σlmνj(k)

δ(ν − νp)δνν′

ω + i0+ + µ− εσkνp

−∆Σα,σp (ω)

[Θα,σlmν′j(k)

]∗]

(B.3)

= − 1

πIm

1

ω + i0+ + µ− εσkνp

−∆Σα,σp (ω)

3∑

j=1

Θα,σlmνpj

(k)[Θα,σlmνpj

(k)]∗ .

If the character m corresponds to an Ir 5d t2g orbital, this expression can be even more simpli-fied. The Wannier functions were indeed constructed from the promoting of the Ir 5d t2g orbital-partuα,σlm,1(r

α) : in the considered energy-window, the one-to-one correspondence between bands and Wan-nier functions can then be extended to the total Ir 5d t2g states. However, since the t2g bands haveboth Ir 5d and O 2p characters,

3∑

j=1

Θα,σlmνpj

(k)[Θα,σlmνpj

(k)]∗

= Dα,σm (k)δmp (B.4)

where Dα,σm (k) is the LDA partial DOS associated to ths pin σ and the m character along the band νp.

As a result, in the energy-window where the t2g Wannier projectors were defined, the spectraldensity Aα,σm (ω) associated to an Ir 5d t2g orbital m can finally be estimated from the following formula:

Aα,σm (ω) = − 1

π

∑

k∈BZ

Im

[Dα,σm (k)

ω + i0+ + µ− εσkνm

−∆Σα,σm (ω)

](5.7)

which is similar to the expression of the spectral function of a one-band model.

B.2 The self energy of a 3/4-filled two-band Hubbard model in theatomic limit

We consider the following two-band Hubbard model with only density-density terms:

H =∑

i=1,2;σ=↑,↓

εic†iσciσ+

∑

i=1,2

Uni↑ni↓+(U −2J)(n1↑n2↓+n2↑n1↓)+(U −3J)(n1↑n2↑+n1↓n2↓) (B.5)

We will assume that the two bands are degenerate (ε1 = ε2 = ε0) and that J ≪ U . As a result, theexpression can be simplified as follows:

H =∑

i=1,2;σ=↑,↓

ε0c†iσciσ +

∑

i=1,2

Uni↑ni↓ + U(n1↑n2↓ + n2↑n1↓) + U(n1↑n2↑ + n1↓n2↓) (B.6)

The states of the system are:

• 1 state |0〉 of energy 0,

• 4 one-particle states |i, σ〉 (i = 1, 2;σ =↑, ↓) of energy ε0,

• 6 two-particle states of energy 2ε0 + U , which we denote |2j〉 with j = 1, .., 6,

• 4 three-particle states |3j〉 with j = 1, ..., 4 of energy 3ε0 + 3U ,

B.2. THE SELF ENERGY OF A 3/4-FILLED TWO-BAND HUBBARD MODEL IN THE ATOMIC LIMIT133

• 1 four-particle state |4〉 of energy 4ε0 + 6U

Our aim is to find the expression of the self-energy of this system when it is 3/4-filled: that is to say,when it accomodates 3 electrons. We perform the calculation at T = 0 K. Because of the degeneracy,the ground-state with three particle is:

|Ψ0〉 =1

4

4∑

j=1

|3j〉 (B.7)

The spectral density of the system is the following:

A(ω) =∑

i=1,2

∑

σ

|〈Ψ0|ciσ|4〉|2δ(ω + µ+ (3ε0 + 3U)− (4ε0 + 6U)) (B.8)

+∑

i=1,2

∑

σ

6∑

j=1

|〈Ψ0|c†iσ|2j〉|2δ(ω + µ+ (2ε0 + U)− (3ε0 + 3U))

=3

4δ(ω + µ− ε0 − 2U) +

1

4δ(ω + µ− ε0 − 3U) (B.9)

As a result, the Green function of the system can be written as:

G(ω) =3

4

1

ω + iη + µ− ε0 − 2U+

1

4

1

ω − iη + µ− ε0 − 3U

=ω + µ− ε0 − 11

4 U

(ω + iη + µ− ε0 − 2U)(ω − iη + µ− ε0 − 3U)(B.10)

And finally, the associated self-energy is:

Σ(ω) = G−10 (ω)−G−1(ω)

= (ω + iη + µ− ε0)−(ω + iη + µ− ε0 − 2U)(ω − iη + µ− ε0 − 3U)

ω + µ− ε0 − 114 U

= (ω + iη + µ− ε0)− (ω + iη + µ− ε0 − 2U)(1 +U4 − iη

ω + µ− ε0 − 114 U

)

= 2U +U

4

ω + iη + µ− ε0 − 2U

ω + µ− ε0 − 114 U

= 2U +U

4

(1 +

34U

ω + µ− ε0 − 114 U

)

= 3

(3

4U

)+

3U2

16

ω + µ− ε0 − 114 U

(B.11)

The chemical potential at T=0 K must lie between ε0 + 2U and ε0 + 3U . We take the value ε0 + 52U

to be more symmetric, which allows to write the following expression for the self-energy:

Σ(ω) = 3

(3

4U

)+

3U2

16

1

ω − U4

(B.12)

Consequently, on the Matsubara axis, the self-energy is:

Σ(iω) =9

4U +

3U2

16

1

iω − U4

= 3

(3

4U

)− 3U3

64

1

ω2 + U2

16

− i3U2

16

ω

ω2 + U2

16

(B.13)

134 APPENDIX B. THE SELF-ENERGY OF SR2IRO4

0 2 4 6 8 10 12-2.25

-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

dxz U=3

dxz U=4

dxz U=5

dxz U=6

model with U=2,3,4,5,6

Figure B.1: Imaginary part of Σdxz(iω) for undistorted Sr2IrO4 without spin-orbit coupling T=300 K(β = 40 eV−1) and with J=0.2 eV. The purple curves are the modelized behavior of the imaginarypart of the self-energy for a 3/4 filled band in the atomic limit. For U≥ 4 eV Σdxz(iω) behaves quitesimilarly as that of the model.

We have used this last expression to fit the self-energy associated to the dxz – or equivalently dyz –orbital for undistorted Sr2IrO4 without the spin-orbit coupling. As observed on figure B.1, the qualityof the fitting increases as U becomes larger, namely when the atomic limit approximation becomesmore and more valid.

To concude, we suggest that the expression (B.13) can be used as a general fitting expression forthe imaginary and real parts of the self-energy each time the system can be considered as a 3/4-filledtwo-band problem.

Appendix C

Structure and conventions in Wien2k

In this appendix, we first describe the structure of a Wien2k calculation in order to complete the generalintroduction to this program which was begun in section 2.2. The second part of this appendix thenfocus on the conventions used in the auxiliary program lapw2 to calculate the coefficients Aναlm(k, σ),Bναlm(k, σ) and Cναlm (k, σ) of equation (2.35). Understanding these conventions was indeed an essential

step in order to develop the interfacing program dmftproj.

C.1 General structure of the Wien2k package

The Wien2k package actually consists of several independent programs. We will give here only a generaldescription of those which are used when a DFT-LDA calculation is performed. More technicalities onthe overall structure of the Wien2k package and on the input and output files can be found in [23].

The initialization

The initialization consists of running a series of small auxiliary programs, which generates the inputsfor the main programs. One first defines the structure of the studied compound – in a file calledcase.struct – and the initialization procedure is then performed by executing the script init_lapw. Itconsists in:

• calculating the muffin-tin radii RαMT for each atom α,

• generating all the symmetry operations of the space group of the compound,

• defining the core, valence and semi-core states,

• generating the k-mesh in the irreducible Brillouin zone,

• calculating finally a starting density ρ(r) for the compound from the superposition of the freeatomic densities.

The self-consistent cycle

The self-consistent cycle is initiated by executing the script run_lapw and is composed of the followingsteps:

• lapw0 which generates the effective Kohn-Sham potential VKS(r) from the density ρ(r) by usingPoisson’s equation to get Vext(r) and the LDA to get V xc(r),

• lapw1 which calculates the eigenvalues εσkν and eigenvectors |ψσkν〉 for the valence electrons,

135

136 APPENDIX C. STRUCTURE AND CONVENTIONS IN WIEN2K

• lapw2 which computes the semi-core and valence densities ρvalence(r) from the eigenvectors,

• lcore which computes core states and their corresponding densities ρcore(r) within a fully rela-tivistic treatment,

• mixer which calculates the total density ρ(r) = ρcore(r)+ρvalence(r) and mixes input and outputdensities

The cycle is repeated until the convergence criterion is met. Usually, the calculation is considered asconverged when the difference in energy between the two last iterations is less than 0.0001 Ry.

Taking into account the spin-orbit coupling

In order to perform a DFT calculation which includes the spin-orbit interaction, the standard procedureconsists in running a regular self-consistent cycle calculation first and then initializing the spin-orbitcoupling parameters for the compound to perform a second self-consistent cycle.

This specific initialization step is done by executing the script initso_lapw. Whereas for a non-spin-polarized case, the generation of the necessary input files is straightforward, care must be takenfor spin-polarized cases: the spin-orbit interaction may indeed lower the symmetry of the system de-pending on how the direction of magnetization is chosen, as explained in [126].

The self-consistent cycle which includes the spin-orbit coupling is then performed by executing thescript run_lapw -so. As described in section 3.3, a second variational approach is used to add thespin-orbit corrections to the scalar-relativistic orbitals calculated in lapw1. As a result, the generalstructure of the self-consistent loop is only slightly modified:

• the program lapwso is called directly after lapw1 to introduce the spin-orbit corrections in thetotal Hamiltonian,

• the basis |ψ+νk〉, |ψ−

νk〉 must be considered (cf. equation (3.52)), which implies that the size of thematrix diagonalized in lapw1 is multiplied by two and the complex version of lapw2 is used .

C.2 Conventions for the symmetry operations in Wien2k

In this section, we introduce how the symmetry operations are stored and calculated in Wien2k. As wewill see in Appendix E, understanding these technicalities are essential to define the projectors from thecoefficients Aναlm(k, σ), B

ναlm(k, σ) and Cναlm (k, σ) of (2.35) and perform the Brillouin zone integration.

C.2.1 Symmetry operations T and local rotations Rloc

As explained in Appendix E, the symmetry operations of the “crystallographic space group” of acompound have the general following form:

∀ r ∈ R3 T (r) = R(r) + v (C.1)

where v is a translation vector which belongs to the Bravais lattice or is a rational fraction of sucha vector and R is an operator of the crystallographic point group of the compound. In Wien2k, thesymmetry operations T of the studied compound are calculated during the initialization step anddescribed at the end of the case.struct file within the following form of a 3× 4 matrix :

rxx rxy rxz txryx ryy ryz tyrzx rzy rzz tz

⇐⇒ ∀ u ∈ R3 T (u) =

rxx rxy rxzryx ryy ryzrzx rzy rzz

uxuyuz

+

txtytz

.

(C.2)

C.2. CONVENTIONS FOR THE SYMMETRY OPERATIONS IN WIEN2K 137

Each matrix is then read in the subroutine init_struct and stored under the following form :

iz(iord) = tR and tau(iord) = t (C.3)

where iord is the number of the symmetry operation. Although iz seems to describe R−1 – we remindthe reader that the matrices R are orthogonal, so that tR = R−1 –, it should be kept in mind thatthis is just a way of storing the information : tiz(iord) will be actually used in most of the programscalled during the calculation.

The previous symmetry operations are defined in the “global coordinate system”. On the contrary,the local orbitals of each atom α of the unit cell are defined in the “ local coordinate system” associatedto this atom α. We insist that this local frame may be different for each atom α, even if they are ofthe same atomic species A. This local frame is defined in two steps:

• First, the symmetry operation T which transforms the first representant of the atomic speciesA into the atom α is applied. This transformation is computed in the subroutine permop of theWien2k package.

• Then, the specific local rotation RAloc associated to the atomic species A and defined in thecase.struct file is applied.

As a result, the transformation from the global coordinate system to the local frame of an atom α is:

Rαloc = RAloc.T . (C.4)

The matrices RAloc are also read in the subroutine init_struct and as previously, tRAloc are actuallystored. In the following, only the operator part R of T will be considered. The translation vectort of T introduce only phase factor in the definition of the coefficients and can thus be neglected forcalculating local quantities.

C.2.2 Representation of the symmetry operations by Euler angles

The symmetry operations T and the local rotation RAloc are described as matrices in real space R3 inthe case.struct file. However, when applied to the coefficient, one must use the corresponding transfor-mation matrices D 1

2 (S) in spin-space or Dl(S) in the space spanned by the spherical harmonics Y lm,

with S = T or RAloc.

In order to generate these matrices, it is necessary to determine if the transformation S is proper(its determinant is 1) or improper (its determinant is −1). In the first case, S can be completelydescribed by three Euler angles α, β and γ. In the second case, S can always be written as the productof an inversion I = −Id and a proper transformation. As a result I.S can be completely described bythree Euler angles α, β and γ.

The three Euler angles associated to a proper transformation S are calculated in the program eulerof the Wien2k package. The convention used is the following:

S = R[z](γ).R[y](β).R[z](α) (C.5)

where R[z](γ) stands for the rotation of angle γ ∈ [0; 2π[ around the z-axis, R[y](β) is the rotationof angle β ∈ [0;π] around the y-axis and R[z](α) the rotation of angle α ∈ [0; 2π[ around the z-axis.


This corresponds in R3 to:

DR3(S) =

cos γ − sin γ 0sin γ cos γ 00 0 1

cosβ 0 sinβ0 1 0

− sinβ 0 cosβ

cosα − sinα 0sinα cosα 00 0 1

=

cos γ cosβ cosα− sin γ sinα − sin γ cosα− cos γ cosβ sinα cos γ sinβsin γ cosβ cosα+ cos γ sinα cos γ cosα− sin γ cosβ sinα sin γ sinβ

− sinβ cosα sinβ sinα cosβ

(C.6)

However, the program euler does not calculate the Euler angles, it rather gives three parameters a,b and c which are linked to α, β and γ as follows:

• if β = 0, then b= 0, c= 0 and a= 2π − α for α ∈]0; 2π[ and a= 0 when α = 0,

• if β = π, then b= 0, a= 0 and c= 2π − γ for γ ∈]0; 2π[ and c= 0 when γ = 0,

• if β ∈]0;π[, then b= β, a= π − α for α ∈ [0;π] and a= 3π − α for α ∈]π; 2π[ and similarly for cand γ.1

C.2.3 Standard computation of the rotation matrices

We remind briefly here how the rotation matrices D 12 (S) or Dl(S) are usually calculated for a proper

rotation, in order to ease the comparison with the matrices computed in Wien2k.

The proper rotations form the O+(3) group which is a compact Lie group with 3 parameters. Letα1, α2 and α3 be these parameters. Then, there exist 3 infinitesimal matrices I1, I2 and I3 such thateach element of this group can be written:

R(α1, α2, α3) = exp I1α1 + I2α2 + I3α3. (C.7)

In particular, for a physical system if 1, 2, 3 stands for x, y, z, it can be shown that:

R(αx, αy, αz) = exp− i

~(lxαx + lyαy + lzαz) (C.8)

where l is the angular momentum of the system. Using the convention used in Wien2k for the Eulerangles (C.5) and the usual relations:

lz|l,m〉 = ~m|l,m〉 and l±|l,m〉 = ~√l(l + 1)−m(m± 1)|l,m± 1〉 with l+ = lx ± ily (C.9)

it is possible to get an explicit form for the irreducible representation associated to the rotation matrixDl(R).

Case of l = 0

For l = 0, the representation is the trivial one :

∀α, β, γ D0(R) = (1). (C.10)

1These relations hold if the matrix used as an input of euler is directly iz(iord).

C.2. CONVENTIONS FOR THE SYMMETRY OPERATIONS IN WIEN2K 139

Case of the spin-space l = 12

For l = 12 , we have the following matrices for the spin moment operators in the basis | ↑〉, | ↓〉:

sx =~

2

(0 11 0

)sy =

~

2

(0 −ii 0

)sz =

~

2

(1 00 −1

). (C.11)

It is therefore quite simple to get the representation of a rotation around the z-axis:

D 12 (R[z](α)) =

(e−

iα2 0

0 eiα2

)D 1

2 (R[z](γ)) =

(e−

iγ2 0

0 eiγ2

)(C.12)

whereas the expression for a rotation around the y-axis is:

D 12 (R[y](β)) =

∞∑

n=0

(−1)n

(2n)!

β2n

22n

(1 00 1

)− i

(−1)n

(2n+ 1)!

β2n+1

22n+1

(0 −ii 0

)=

(cos β2 − sin β

2

sin β2 cos β2

)

(C.13)

Finally, we got the following general expression:

D 12 (R) =

(cos β2 e

−iα+γ2 − sin β

2 eiα−γ

2

sin β2 e

−iα−γ2 cos β2 e

iα+γ2

)(C.14)

when R is defined by the Euler angles α, β and γ.

Case of orbital-spaces with l ≥ 1

For integer values of l ≥ 1, the representation needs of course more computational effort. We will onlyperform the calculation in the case l = 1. We have the following matrices for the kinetic operators inthe basis |1,−1〉, |1, 0〉, |1, 1〉:

lx =~√2

0 1 01 0 10 1 0

ly =

i~√2

0 1 0−1 0 10 −1 0

lz = ~

−1 0 00 0 00 0 1

. (C.15)

The representation of a rotation around the z-axis are thus:

Dl(R[z](α)) =

eiα 0 00 1 00 0 e−iα

Dl(R[z](γ)) =

eiγ 0 00 1 00 0 e−iγ

(C.16)

whereas the expression for a rotation around the y-axis is:

Dl(R[y](β)) = I3 +∞∑

n=1

1

(2n)!

β2n

2n

−(−2)n−1 0 (−2)n−1

0 (−2)n 0(−2)n−1 0 −(−2)n−1

+∞∑

n=0

1

(2n+ 1)!

β2n+1

2n√2

0 (−2)n 0−(−2)n 0 (−2)n

0 −(−2)n 0

=

1 + cosβ2

sinβ√2

1− cosβ2

−sinβ√2

cosβsinβ√

21− cosβ

2 −sinβ√2

1 + cosβ2

. (C.17)


Finally, the general expression of a rotation matrix is:

D1(R) =

1 + cosβ2 ei(α+γ)

sinβ√2eiγ

1− cosβ2 e−i(α−γ)

−sinβ√2eiα cosβ

sinβ√2e−iα

1− cosβ2 ei(α−γ) −sinβ√

2e−iγ

1 + cosβ2 e−i(α+γ)

(C.18)

when R is defined by the Euler angles α, β and γ.

C.2.4 Computation of the rotation matrices in Wien2k

A spinor rotation D 12 (S) is defined by the following matrix in the basis | ↑〉, | ↓〉 :

∀α, β, γ D 12 (a, b, c) =

(cos b

2ei a+c

2 sin b

2e−i a−c

2

− sin b

2ei a−c

2 cos b

2e−i a+c

2

). (C.19)

The rotation matrices Dl(S) are calculated by the program dmat which creates the matrix whosecoefficients are in the basis |l,−l〉, . . . , |l, l〉:

(Dl(a, b, c)

)m,n

= ei(na+mc)

√(l +m)!(l −m)!

(l + n)!(l − n)!.

2l∑

k=0

(−1)l−m−kC l,km,n(sinb

2)2l−m−n−2k(cos

b

2)2k+m+n

with C l,km,n =(l + n)!(l − n)!

(l −m− k)!(m+ n+ k)!(l − n− k)!k!and m,n ∈ [−l; l] (C.20)

For each(Dl(a, b, c)

)m,n

, the value of k considered in the sum are only those for which :

l −m− k ≥ 0 and l − n− k ≥ 0 and m+ n+ k ≥ 0. (C.21)

To understand better this expression, we compute explicitly the rotation matrix in the subspacel = 1. The basis considered is thus |1,−1〉, |1, 0〉, |1, 1〉 and we get:

D1(a, b, c) =

(cos b

2)2e−i(a+c) −

√2 sin b

2 cosb

2e−ic (sin b

2)2ei(a−c)

√2 sin b

2 cosb

2e−ia (cos b

2)2 − (sin b

2)2 −

√2 sin b

2 cosb

2eia

(sin b

2)2e−i(a−c)

√2 sin b

2 cosb

2eic (cos b

2)2ei(a+c)

(C.22)

=

1 + cos b2 e−i(a+c) −sin b√

2e−ic 1− cos b

2 ei(a−c)

sin b√2e−ia cos b −sin b√

2eia

1− cos b2 e−i(a−c) sin b√

2eic 1 + cos b

2 ei(a+c)

By comparing this expression with (C.18) and by taking into account the definition of the param-eters a, b and c, it appears that D1(a, b, c) corresponds to Dl(R). In spin-space however, some phasefactors need to be compensated because of factor 1

2 .

We have only discussed here the proper transformation. As we mentioned previously, a transfor-mation can also be improper and can thus be written as the product of a proper rotation and a spatialinversion I = −Id. However, inversion I acts only on the r-space and doesn’t affect the spin-space.As a result, taking into account this transformation consists merely in adding a factor (−1)l in Dl(R)

– and Dl(a, b, c) – for integer values of l, whereas no change in D 12 (R) – and D 1

2 (a, b, c) – occurs.

C.3. CONVENTIONS FOR THE COEFFICIENTS AναLM (K, σ), BναLM (K, σ) AND CναLM (K, σ) 141

C.3 Conventions for the coefficients Aναlm(k, σ), B

ναlm(k, σ) and Cνα

lm(k, σ)

The charge density ρ(r) is calculated in lapw2 in the global coordinate system of the compound. How-ever, the coefficients Aναlm(k, σ), B

ναlm(k, σ) and Cναlm (k, σ) of (2.35) are defined in the local coordinate

system associated to the atom α. A multiplication by a transformation from the global frame to thelocal frame of the atom α is then applied in the program lapw2.

However, contrary to what we would have expected, we noticed that this transformation is appliedwithout taking into account the spinor rotation matrix and even when the calculation explicitly includesthe spin-orbit coupling. In order to remain consistent in our implementation of dmftproj, we thus hadto introduce this lacking coefficients in the definition of the coefficients Aναlm(k, σ), B

ναlm(k, σ) and

Cναlm (k, σ). This is done in the program set_projections.

Appendix D

Description of dmftproj

In this appendix, we present the general structure of the interfacing program dmftproj and give a brieftutorial to its use. More information are available in [110].

D.1 General structure of dmftproj

The dmftproj program is structured as follows:

• the input file case.indmftpr is first read in order to define which projectors Pασlm,ν(k) and Θα,σlmνj(k)

will be calculated during the execution.

• the Θαlmνj(k) projectors are then built without any restriction in the energy window, in order

to evaluate the density matrices of each orbital character. The results obtained are the same asobtained with the program lapwdm of the Wien2k package.

• the projectors Pασlm,ν(k) are then constructed for the chosen orbitals – the “correlated” ones – andthe density matrix associated to the obtained Wannier orbitals is finally displayed.

The dmftproj program is composed of the following auxiliary subroutines:

set_ang_trans: it sets the transformation from the complex basis to the desired one.

setsym: it sets the symmetry matrices of the system after reading the case.dmftsym file.

set_rotloc: it sets the local rotation matrices Rαloc for each atom of the system (cf. Appendix C).

set_projections: it computes the projectors Pασlm,ν(k) and Θαlmνj(k), as explained in sections 2.3, 3.3

and 3.4.

orthogonal_wannier : it performs the orthogonalization for the projectors Pασlm,ν(k) in order to getthe final Pασlm,ν(k) projectors.

density: it calculates the local density matrix associated to each atom.

symmetrize_mat: it performs the symmetrization in order to get the integration over the wholeBrillouin zone when the density matrices are calculated. (cf. discussion in section 2.3 andAppendix E)

To allow the interfacing, some changes were also introduced in the program lapw2 in order to getthe coefficients Aναlm(k, σ), B

ναlm(k, σ) and Cναlm (k, σ) of (2.35) in the file case.almblm and the description

of the symmetry operations in the file case.dmftsym. However, we will not describe the technicalitiesof these modifications here.

143

144 APPENDIX D. DESCRIPTION OF DMFTPROJ

D.2 Description of the master input file case.indmftpr

The master input file case.indmftpr of dmftproj has the following structure. We give an example of sucha file in the case of Sr2IrO4.

3 ! nsorta) 1 2 2 2 ! mult(isrt)

3 ! lmaxfromfile ! basisfile_name1 1 2 0 ! lsort(l)

b-1) 0 0 2 0 ! nb_irep(l)010 ! ifSOcomplex ! basis

b-2) 1 0 0 0 ! lsort(l)0 0 0 0 ! nb_irep(l)cubic ! basis1 1 2 0 ! lsort(l)

b-3) 0 0 1 0 ! nb_irep(l)0 ! ifSO

c) -0.26 0.48 ! e_bot and e_top0.65650 ! e_Fermi

Part a) is the general description for the considered material:

nsort: number (integer) of inequivalent atomic sort in the system,

mult(isrt): nsort integers equal to the multiplicity of each sort in the unit cell,

lmax: maximal l number (integer) considered in the system.

Part b) describes the chosen “treatment” for each atomic sort (it must then be repeated nsort time).Moreover, the order for the description must be the same as in the case.struct file.

basis: name associated to the description of the basis (for the projector). It can be:

• complex for complex spherical harmonics,

• cubic for cubic spherical harmonics.

• fromfile for a basis described in an added file. In this last case, the complete name of thefile must be written the line after. In the example, the file is thus called “file_name”. Thisfile must be in the directory where the computation is performed. It must contain thedescription of the basis for all the included orbital. No newline for the separation of thebasis description is needed. The s-orbital does not need a basis description in this file, if itis included.

lsort(l): (lmax+1) integers to describe the “treatment” of each orbital:

• 0 means that the corresponding orbital is not included,

• 1 means that the orbital is included but no projector Pασlm,ν(k) will be calculated for it. (thiscorresponds to an orbital “significantly present” in the considered energy window but whichdo not require a DMFT calculation to treat its correlation level),

D.3. EXECUTION OF THE PROGRAM 145

• 2 means that the orbital is included and a projector Pασlm,ν(k) will be calculated for it. (thistypically corresponds to an orbital “significantly present” in the considered energy windowand correlated.)

nb_irep(l): (lmax+1) integers to describe the number of irreducible representation (irep) to considerfor each orbital l. This description is read (and used) only if the corresponding lsort(l)=2. onlyone particular irep can be “correlated ”. If the number of irep is more than 2 for one orbital, theline after must describe the treatment of each irep (nb_irep(l) number in this line) with a flag(0/1) (cf. case b-1):

• 0 means that the orbital is not correlated ; no projector Pασlm,ν(k) will be calculated for it,

• 1 means that the orbital is correlated and a projector Pασlm,ν(k) will be calculated for it.

ifSO: flag (0/1) which states if the spin-orbit coupling is taken into account for this atomic sort. Thisflag is necessary only if one of the orbital of the sort is correlated. That is why this line does notappear in the section b-2.

Part c) finally precises the last options for the computation:

e_bot and e_top define the size of the energy window (given in Rydberg and relatively to the Fermilevel),

e_Fermi is the Fermi level obtained at the end of the DFT calculation. This line is read only whenone wants to calculate the momentum-resolved spectral function for the system.

Remark : The number of irreducible representation depends on the chosen basis description.For the complex basis, none is possible. (irep=0).For the cubic basis, they are implemented only in the case without any spin-orbit coupling (for

instance : eg-t2g for l = 2).When one uses the option fromfile, several irep could be considered even with spin-orbit coupling

but it depends of course on the chosen basis. In the added file, the irep are indicated by a star (*) atthe beginning of a line.

D.3 Execution of the program

D.3.1 In order to perform a DMFT calculation

Before running dmftproj, the coefficients Aναlm(k, σ), Bναlm(k, σ) and Cναlm (k, σ) of (2.35) and some other

general informations on the system (symmetry operations and local rotations) must be calculated. Thelapw2 program has thus to be executed in the following way:

x lapw2 -alm [-up/-dn -c -so]

The new flag “-alm” is written in the first line of case.in2(c) so that the input files (case.almblm(up/dn)and case.dmftsym) for dmftproj are produced. The master input file required by dmftproj is calledcase.indmftpr and was described previously. The program should be executed as follows:

dmftproj [-sp] [-so]

The options stand for spin-polarized (-sp) calculations and for calculation which takes into accountthe spin-orbit interaction (-so). For technical reasons, the spin-orbit coupling is however implementedfor the spin-polarized case only.

146 APPENDIX D. DESCRIPTION OF DMFTPROJ

We remind that up to now the dmftproj program can be used only with a Continuous-Time QuantumMonte Carlo (CTQMC) impurity solver and an Hubbard-I solver.

D.3.2 In order to calculate a momentum-resolved spectral function

To produce the output needed to calculate the momentum-resolved spectral function after a DMFT cal-culation was performed, one must first prepare the desired k-point path for A(k, ω) in a case.klist_bandfile. Then, the same procedure as used in Wien2k to calculate Kohn-Sham band structure must beapplied:

x lapw1 [-up/-dn-so] -bandx lapw2 -alm [-up/-dn-so] -band

After putting the Fermi energy of the system in the last line of the file case.indmftpr, it is thenenough to execute the following command in the same directory:

dmftproj [-sp] [-so] -band

D.4 Description of the output files

D.4.1 Output to perform a DMFT calculation

The subroutine outputqmc.f of dmftproj produces the following four data files required to perform aDMFT calculation and some simple post-processing of the results:

i) case.ctqmcout and case.symqmc: these files contain all the main data to perform the DMFTself-consistent loop.

More precisely, case.ctqmcout describes the number of k-points in IBZ, the total number of or-bitals, and correlated orbitals, the multiplicity of each sort of atoms, the rotation matrices, theprojectors Pασlm,ν(k) to correlated orbitals, the k-point weights, the eigenvalues of the Hamilto-nian in Kohn-Sham basis. case.symqmc contains the number of symmetry operations for thesymmetrization, the corresponding permutation matrices, the rotations matrices Dl

mm′(R), andadditional transformation if symmetry-adapted basis is required

ii) case.parproj and case.sympar: thes files contain the same informations as the previous ones butare used to recalculate partial quantities (for example the spectral density) after the DMFTcalculation. Instead of describing the Wannier projectors, they give the projectors Θα,σ

lmνj(k).

D.4.2 Output for calculating a momentum-resolved spectral function

The flag -band allows to produce the file case.outband which contains: the projectors of the correlatedorbitals, Pασlm,ν(k) for the new k-points, the corresponding eigenvalues, the projectors Θα

lmνj(k), if somefatband character plotting are desired, information on the new k-list (especially, the labels for specialk-points).

It is then possible to plot the momentum-resolved spectral function, to compare the result of theLDA+DMFT calculation with the experimental curves obtained with Angle-Resolved Photo-EmissionSpectroscopy (ARPES).

More precise description of these files can be found in the tutorial of the programm [110].

Appendix E

Symmetry operations and projectors

In this appendix, we explain the formula to integrate over the all Brillouin zone (for local quantitieslike Gloc, the density matrix or the spectral density A(ω)). As we have highlighted at the end of thesection XXX, we have to go from the irreducible BZ to the full BZ and then use symmetry operationof the space group of the compound. However, in Wien2k and dmftproj, this is not the ordinary“crystallographic space group” which is considered in this sum but the “Shubnikov (magnetic) group”of the compound.

In a first part, we give an introduction to the “Shubnikov (magnetic) group” which introducethe time-operator Θ. we then remind the unusual properties of an anti-symmetry operators, beforedetermining the modification of the projectors when they are applied any symmetry operations. Weconclude by deriving the equations...

E.1 Space groups and time reversal operator

As mentioned in section 1.1, to describe a crystal – in which the atomic nuclei are arranged in anorderly periodic pattern in the three spatial dimensions –, it is convenient to define the “Bravaislattice” B and the “crystallographic point group” of the compound. More generally, one introduces the“crystallographic space group” of a crystal as the set of all the symmetry operations T which leave thesystem invariant after applying any of them to it. These operations have the following form:

∀ r ∈ R3 T (r) = R(r) + v (E.1)

where v is a translation vector which belongs to the Bravais lattice or is a rational fraction of such avector and R is an operator of the crystallographic point group: a rotation through integral multipleof 2π

n about some axis, a rotation-reflection, a rotation-inversion, a reflection or an inversion.By combining the 32 crystallographic point group with the 14 Bravais lattices, there turn out to be

230 different space groups in three dimensions. More details can be found in [16, 25]. These groupswere regarded as the ultimate development in the study of the symmetry of a crystal until Shubnikovintroduced the idea of operations of anti-symmetry in 1951 [147].

Brief introduction to Shubnikov magnetic groups

By considering an extra coordinate, with only two possible values, in addition to the ordinary positioncoordinates – for instance, the direction of a local magnetic moment, parallel or anti-parallel to a givendirection –, it is possible to define the operation of anti-symmetry as the operation which changes thevalue of this coordinate. If we call this operation Θ, it is then possible to have compound operationsof anti-symmetry corresponding to the performance of both an ordinary space group operation T – orpoint group operation – together with the operation Θ.

147

148 APPENDIX E. SYMMETRY OPERATIONS AND PROJECTORS

As a result, many more point groups and space groups can be defined. They are respectively 122and 1651 of them and they are referred to the “Shubnikov (magnetic) groups”.

The Shubnikov groups are of great use to describe the symmetry of magnetic crystals. In this case,Θ is the operation which reverses the magnetic moment and corresponds to the time reversal operation.A change of magnetic moment can indeed be thought as caused by a reversal of the direction of theelectric current which gives rise to it and this reversal of the direction is equivalent to a reversal of thesense of the direction of the time variable (since i = dq

dt ).

Classification of Shubnikov point groups

In the following, we will only consider Shubnikov magnetic point groups. If G denotes one of theordinary crystallographic point groups, there are three types of Shubnikov group corresponding to it:

type I: the “uncolored ” or standard point group G;

type II: the “grey” point group defined as G +ΘG

type III: the “black and white” point group which is given by: H+Θ(G rH), where H is a halvingsubgroup of G.

Type II or grey point groups possess the operation of time inversion Θ itself as a symmetry op-eration. As a result, no spontaneous magnetic field can exist anywhere in the crystal. Therefore,either the individual atoms within the crystal have no magnetic moments – the compound can onlyexhibit diamagnetism –, or they do possess spontaneous magnetic moments but randomly oriented –the compound is paramagnetic –.

On the contrary, the operation of antisymmetry is absent in type I groups or is only present incombination with a symmetry operation R in type III groups. Consequently, it may be possible for acrystal described by these groups to possess a net magnetic moment – ferromagnetic or ferrimagneticcrystal – or to be antiferromagnetic.

We will not going further in the theoretical description of Shubnikov magnetic point groups andtheir properties. The interested reader can find a detailled presentation in [25].

In Wien2k, a “grey” (type II) point group is always used when performing a paramagnetic com-putation and a “black and white” (type III) point group is always considered when the calculationis spin-polarized. The same conventions were applied when implementing the interfacing programdmftproj.

E.2 Properties of an antilinear operator

Any anti-symmetry – and among them the time reversal operation – is described by an antilinearoperator Θ:

∀λ, µ ∈ C ∀u,v Θ(λu+ µv) = λ∗Θ(u) + µ∗Θ(v) (E.2)

and more precisely an antiunitary operator – with the property Θ† = Θ−1. In this section, we brieflyremind some unusual properties of such operators.

General properties

Let Θ1 and Θ2 be two operators.If both Θ1 and Θ2 are antilinear, their product Θ1Θ2 is a linear operator. If Θ1 – or Θ2 – is

antilinear and the other is linear, the product Θ1Θ2 is still antilinear.If Θ1 is an antilinear operator, its inverse Θ−1

1 and its Hermitian conjugate Θ†1 are antilinear too.

E.2. PROPERTIES OF AN ANTILINEAR OPERATOR 149

The time reversal operator Θ and the conjugation operator K

In the basis |r, σ〉, the expression of the time reversal operator Θ is:

Θ = −iσy.Kr with σy =

(0 −ii 0

)(E.3)

where Kr is the conjugation operator in this same basis, or in other words:

∀|r, σ〉 ∀λ Kr (λ|r, σ〉) = λ∗|r, σ〉 (E.4)

More generally, it can be shown [112] that any antiunitary operator can be written as the product ofthe complex conjugation operator K times a linear “standard” operator:

Θ = ΘK2 = (ΘK)K = K(KΘ) since K2 = Id (E.5)

Matrix representation of an antilinear operator

It is possible to use a matrix representation for K or any antilinear operator. In this case, it is necessaryto impose the following convention: all the matrices at the right-hand side of the “antilinear” matrixshould be complex conjugate.

Indeed, let (ei)i∈I be the basis of a C-vector space and T = D(Θ) the representation of the operatorΘ in this vector space. Moreover, let B = D(f) be the representation of the endomorphism f .

∀j ∈ I Θ.f(ej) = Θ(∑

k

bkjek) =∑

k

b∗kjΘ(ek) =∑

i,k

tikb∗kjei ⇐⇒ D(Θ.f) = T.B∗ (E.6)

∀j ∈ I f.Θ(ej) = f(∑

k

tkjek) =∑

k

tkjf(ek) =∑

i,k

biktkjei ⇐⇒ D(f.Θ) = B.T (E.7)

If f is antilinear too, it can be shown in a similar way that:

D(Θ.f) = T.B∗ and D(f.Θ) = B.T ∗ (E.8)

As a result, in the particular case where Θ is bijective and f = Θ−1, we have the following relation:

T.(T−1)∗ = T−1.T ∗ = Id (E.9)

Change of basis

As seen in (E.4), the definition of the K operator depends explicitly of the basis. This property holdsalso for any antilinear operator.

Let’s now consider that f is an automorphism, that is to say a bijective endomorphism. Then iffi = f(ei), (fi)i∈I is also a basis of our considered C-vector space. Let B = Dei(f) and C = B−1 =Dfi(f

−1). We are now interested in the matrix representation of Θ in this new basis.

∀j ∈ I Θ(f(ej)) = Θ(∑

l

bljel) =∑

k

b∗ljΘ(el) =∑

k,l

tklb∗ljek =

∑

i,k,l

ciktklb∗ljf(ei) (E.10)

and therefore

Dfi(Θ) = C.Dei(Θ).B∗ or Tfi = B−1.Tei .B∗ (E.11)


Hermitian adjoint of an antilinear operator

The usual definition – for a linear operator A – of its Hermitian adjoint is:

〈A†y, x〉 = 〈y,Ax〉 (E.12)

In the case of an antilinear operator Θ, this definition is slightly modified:1

〈Θ†y, x〉 = (〈y,Θx〉)∗ or 〈x,Θ†y〉 = 〈y,Θx〉 (E.13)

Let then be T = D(Θ) and U = D(Θ†). The usual relation for linear operators – namely U = tT ∗

– doesn’t hold anymore for antilinear operator:

∀ i, j ∈ I 〈ei,Θ†ej〉 = 〈ei,∑

k

ukjek〉 =∑

k

ukj〈ei, ek〉 = uij

〈ei,Θ†ej〉 = 〈ej ,Θei〉 = 〈ej ,∑

k

tkiek〉 =∑

k

tki〈ej , ek〉 = tji (E.14)

Therefore the true relation is merely: U = tT

E.3 Action of a space group operation on the projectors

In order to obtain the formulas involving a sum over the irreducible Brillouin zone, the action of thespace group operations on the Wannier and Θ−projectors, Pα,σlm,ν(k) and Θα,σ

lm,νj(k), must be known.In this section, we calculate it explicitly.

In the following, we will only consider the ordinary point group operators2 R and we will denoteΘ the time reversal operator. According to section E.1, we thus have to study symmetry operations Sof the form R or ΘR

E.3.1 Action of a space group operation on the Bloch states

From Bloch theorem, we have:

Hψk,ν(r) = εk,νψkν(r) ⇐⇒ ψk,ν(r) = eik.ruk(r) with uk,ν(r+R) = uk,ν(r) ∀ R ∈ B(E.15)

By definition, the symmetry operators S are such that [H,S] = 0. Consequently, if |ψkν〉 is an eigen-vector of H, S|ψkν〉 is also an eigenvector of H with the same eigenvalue. We now precise the action ofS on the momentum k and the spin index σ. We will not study the effect of the symmetry operationon the band index ν, since it really depends on what it is standing for. We will merely denote Sν asthe image by S of ν in the following.

If S = R,〈r|S|ψk,ν〉 = 〈R−1r|ψk,ν〉 = ψkν(R−1r) = eik·R

−1ruk,ν(R−1r) (E.16)

Defining uRk,Rν(r) = uk,ν(R−1r) which still verifies uRk,Rν(r+R) = uRk,Rν(r) ∀ R ∈ B, we can thenwrite:

〈r|S|ψk,ν〉 = eiSk·ruRk,Rν(r) = ψSk,Sν(r) = 〈r|ψSk,Sν〉 hence S|ψk,ν〉 = |ψSk,Sν〉 (E.17)

1The existence of the Hermitian adjoint is shown by using the Riesz representation theorem. In order to apply it, theconjugate expression of the scalar product must indeed be considered.

2Taking into account the translation vector v of the more general space group operation T only introduce a phasefactor, which will anyway disappear when local quantities are calculated.

E.3. ACTION OF A SPACE GROUP OPERATION ON THE PROJECTORS 151

If we add the spin quantum number, we have to consider the spinor |Ψk〉 and the representation of thetransformation S in the spin-space, D

12 (S), introduced in Appendix C:

|Ψk,ν〉 =(

|ψ↑k,ν〉

|ψ↓k,ν〉

)and S|Ψk,ν〉 = D 1

2 (S)|ΨSk,Sν〉 ⇔ S|ψσk,ν〉 =∑

τ=↑,↓

D 12 (S)τσ|ψτSk,Sν〉

(E.18)The result is similar if the computation includes the spin-orbit coupling:

S|Ψk,ν〉 = D 12 (S)|ΨSk,Sν〉 with |Ψk,ν〉 =

(|ψ+

k,ν〉|ψ−

k,ν〉

)⇔ S|ψik,ν〉 =

∑

j=+,−

D 12 (S)ji|ψτSk,Sν〉 (E.19)

Before studying the case S = ΘR, we consider the action of the time-reversal operator Θ alone ona Bloch state. Neglecting first the spin, we get:

Θ|ψk,ν〉 = Θ

(∑

r

eik·ruk,ν(r)|r〉)

=∑

r

e−ik·ru∗k,ν(r)|r〉 (E.20)

since Θ|r〉 = |r〉. Moreover, defining u−k,Θν(r) = u∗k,ν(r), we can finally write:

Θ|ψk,ν〉 =∑

r

ei(−k)·ru−k,Θν(r)|r〉 = |ψ−k,Θν〉 (E.21)

Then, if we add the spin degree of freedom, the expression becomes:

Θ|Ψk,ν〉 = −iσy.Kr

(|ψ↑

k,ν〉|ψ↓

k,ν〉

)= −iσy

(|ψ↑

−k,Θν〉|ψ↓

−k,Θν〉

)=

(−|ψ↓

−k,Θν〉|ψ↑

−k,Θν〉

)(E.22)

and the result is similar if the spin-orbit coupling is included – it is then enough to replace ↑, ↓ by +,−respecitively.

If S = ΘR, the result is merely derived from the previous relations:

S|ψσk,ν〉 = Θ

(∑

τ

D 12 (R)τσ|ψτRk,Rν〉

)=∑

τ

D 12 (R)∗τσΘ|ψτRk,Rν〉 =

∑

τ,µ

(−iσy)µτD12 (R)∗τσ|ψµ−Rk,Sν〉

(E.23)Taking into account the following relations:

Sk = −Rk and D 12 (ΘR) = D 1

2 (Θ)D 12 (R)∗ with D 1

2 (Θ) = −iσy, (E.24)

we finally can write:S|ψσk,ν〉 =

∑

τ

D 12 (ΘR)τσ|ψτSk,Sν〉,

which is analogous to the formulas (E.18) and (E.19).

E.3.2 Transformation of the projectors under a space group operation

case of a symmetry operation S = RLet’s assume first of all that S is a simple space group operation (S = R). As defined in section 2.3,the temporary Wannier projectors are written as follows:

Pα,σlm,ν(k) = 〈uα,σl (Eα1l)Ylm|ψσkν〉, ∀ ν ∈ W (E.25)


Pα,σlm,ν(Sk) = 〈uα,σl (Eα1l)Ylm|ψσSk,ν〉 = 〈uα,σl (Eα1l)Y

lm|ψS(S−1σ)

Sk,S(S−1ν)〉

= 〈uα,σl (Eα1l)Ylm|(S|ψS−1σ

k,S−1ν〉)

=[〈ψS−1σ

k,S−1ν |(S−1|uα,σl (Eα1l)Y

lm〉)]∗

since S−1 = S†

=∑

m′

Dl(S−1)∗m′,m 〈ψS−1σk,S−1ν |u

S−1α,S−1σl (ES−1α

1l )Y lm′〉∗

=∑

m′

Dl(S)m,m′ 〈uS−1α,S−1σl (ES−1α

1l )Y lm′ |ψS−1σ

k,S−1ν〉 (E.26)

because Dl(S−1) = Dl(S†) = Dl(S)† = [tDl(S)]∗. This finally leads to:

Pα,σlm,ν(Sk) =∑

m′

Dl(S)m,m′ PS−1α,S−1σlm′,S−1ν

(k) (E.27)

It can then be shown that:

Oα,α′

m,m′(Sk, σ) =∑

q,q′

Dl(S)m,q OS−1α,S−1α′

q,q′ (k,S−1σ) Dl(S−1)q′,m′ (E.28)

[O−1/2(Sk, σ)

]α,α′

m,m′=

∑

q,q′

Dl(S)m,q[O−1/2(k,S−1σ)

]S−1α,S−1α′

q,q′Dl(S−1)q′,m′ (E.29)

and thus the same relation as (E.27) also for the "true" projectors Pα,σlm,ν(Sk):


m′

Dl(S)m,m′ PS−1α,S−1σlm′,S−1ν

(k) (E.30)

Moreover, when the calculation is paramagnetic – up and down states are thus degenerate –, the ex-pression (E.30) can be used without paying attention to the indices σ and S−1σ.

When the spin-orbit coupling is taken into account, the temporary Wannier projectors are writtenas follows:

[Pα,σlm,ν

]i(k) = 〈uα,σl (Eα1l)Y

lm|ψikν〉, ∀ ν ∈ W and with i = +,−. (E.31)

The relation (E.27) then becomes:

[Pα,σlm,ν

]i(Sk) =

∑

m′

Dl(S)m,m′

[PS−1α,S−1σlm′,ν

]S−1(i)(k)

=∑

m′,τ,j

Dl(S)m,m′ D 12 (S)σ,τ D 1

2 (S−1)j,i 〈uS−1α,τl (ES−1α

1l )Y lm′ |ψ j

k,S−1ν〉

=∑

m′,τ,j

Dl(S)m,m′ D 12 (S)σ,τ

[PS−1α,τlm′,ν

]j(k) D 1

2 (S−1)j,i (E.32)

which leads to:[Pα,σlm,ν

]i(Sk) =

∑

m′,τ,j


[PS−1α,τlm′,ν

]j(k) D 1

2 (S−1)j,i (E.33)

E.3. ACTION OF A SPACE GROUP OPERATION ON THE PROJECTORS 153

case of an antisymmetry operation S = ΘR

Starting from the defintion of the temporary Wannier projectors (E.25), we get:

Pα,σlm,ν(Sk) = 〈uα,σl (Eα1l)Ylm|ψσSk,ν〉 = 〈uα,σl (Eα1l)Y

lm|ψS(S−1σ)

Sk,S(S−1ν)〉

= 〈uα,σl (Eα1l)Ylm|(S|ψS−1σ

k,S−1ν〉)

= 〈ψS−1σk,S−1ν |

(S−1|uα,σl (Eα1l)Y

lm〉)

since S−1 = S†

=∑

m′

Dl(S−1)m′,m 〈ψS−1σk,S−1ν |u

S−1α,S−1σl (ES−1α

1l )Y lm′〉

=∑

m′

Dl(S)m,m′ 〈uS−1α,S−1σl (ES−1α

1l )Y lm′ |ψS−1σ

k,S−1ν〉∗ (E.34)

because Dl(S−1) = Dl(S†) = Dl(S)† = tDl(S). This finally leads to:


m′

Dl(S)m,m′

[PS−1α,S−1σlm′,S−1ν

(k)]∗

(E.35)


Oα,α′

m,m′(Sk, σ) =∑

q,q′

Dl(S)m,q[OS−1α,S−1α′

q,q′ (k,S−1σ)]∗ [

Dl(S−1)q′,m′

]∗(E.36)

[O−1/2(Sk, σ)

]α,α′

m,m′=

∑

q,q′

Dl(S)m,q([O−1/2(k,S−1σ)

]S−1α,S−1α′

q,q′Dl(S−1)q′,m′

)∗

(E.37)

and thus the same relation as (E.35) also for the "true" projectors Pα,σlm,ν(Sk):


m′

Dl(S)m,m′

[PS−1α,S−1σlm′,S−1ν

(k)]∗

(E.38)

Moreover, when the calculation is paramagnetic – up and down states are thus degenerate –, the ex-pression (E.30) can be used without paying attention to the indices σ and S−1σ.

When the spin-orbit coupling is taken into account, the relation (E.35) then becomes:

[Pα,σlm,ν

]i(Sk) =

∑

m′

Dl(S)m,m′

([PS−1α,S−1σlm′,ν

]S−1(i)(k)

)∗

=∑

m′,τ,j


[D 1

2 (S−1)j,i

]∗〈uS−1α,τl (ES−1α

1l )Y lm′ |ψ j

k,S−1ν〉∗

=∑

m′,τ,j


[[PS−1α,τlm′,ν

]j(k)

]∗ [D 1

2 (S−1)j,i

]∗(E.39)

which leads to:

[Pα,σlm,ν

]i(Sk) =

∑

m′,τ,j


[[PS−1α,τlm′,ν

]j(k)

]∗ [D 1

2 (S−1)j,i

]∗(E.40)


E.4 Local quantities and sum over the irreducible Brillouin zone:

We consider in this section a quantity O which has the following form in the Bloch basis |ψσkν〉:

〈ψσkν |O|ψσ′

k′ν′〉 = Oσσ′

νν′ (k)δkk′ , (E.41)

and which commutes with any symmetry operation of the system [O,S] = 0. This is particularly thecase of the Hamiltonian H, the Green function G, the charge density ρ.

We would like to compute the value of the elements (Oαl )σσ′

mm′ in the local basis |wασlm〉 of Wannierfunctions:

(Oαl )σσ′

mm′ = 〈wασlm |O|wασ′

lm′ 〉 (E.42)

E.4.1 Case of a paramagnetic compound

without spin-orbit coupling

If we consider a paramagnetic compound – with a “grey” (type II) magnetic point group G +ΘG –, wehave to perform the following computation:

(Oαl )σσ′

mm′ = 〈wασlm |

∑

k∈1BZ

∑

ν,ν′

|ψσkν〉Oσσ′

νν′ (k)〈ψσ′

kν′ |

|wασ′

lm′ 〉

=∑

k∈1BZ

∑

ν,ν′

Pα,σlm,ν(k) Oσσ′

νν′ (k)[Pα,σ

′

lm′,ν′(k)]∗. δσσ′ (E.43)

where 1BZ is the first Brillouin zone of the original point group G. In the following, we will note(Oα,σl

)mm′ = (Oα

l )σσmm′ and Oσ

νν′(k) = Oσσνν′(k).

(Oα,σl

)mm′ =

∑

S∈G

∑

k∈IBZ

∑

ν,ν′

Pα,σlm,ν(Sk) Oσνν′(k)

[Pα,σlm′,ν′(Sk)

]∗

=1

2

∑

S∈G

∑

k∈IBZ

∑

ν,ν′

Pα,σlm,ν(Sk) Oσνν′(Sk)


]∗

+1

2

∑

S∈G

∑

k∈IBZ

∑

ν,ν′

Pα,σlm,ν(−ΘSk) Oσνν′(−ΘSk)

[Pα,σlm′,ν′(−ΘSk)

]∗

=1

2

∑

S∈G

∑

k∈IBZ

∑

ν,ν′

Pα,σlm,ν(Sk) Oσσ′

νν′ (Sk)[Pα,σlm′,ν′(Sk)

]∗

+1

2

∑

S∈ΘG

∑

k∈IBZ

∑

ν,ν′



]∗

(E.44)

where IBZ denotes the irreducible Brillouin zone of the original point group G. Moreover, since we canalways choose a Brillouin zone with k = 0 as center of symmetry, it is possible not to take into accountthe introduced sign of the k point in the second right-hand-side term. The two terms in (E.43) can berewritten as follows:

E.4. LOCAL QUANTITIES AND SUM OVER THE IRREDUCIBLE BRILLOUIN ZONE: 155

• when S ∈ G:

[(Oα,σl

)mm′

]G=

∑

S∈G

∑

k∈IBZ

∑

ν,ν′



]∗

=∑

S∈G

∑

n,n′

Dl(S)m,n[(

OS−1α,S−1σl

)nn′

]unsym

Dl(S−1)n′,m′

with[(Oα,σl

)mm′

]unsym

=∑

k∈IBZ

∑

ν,ν′

Pα,σlm,ν(k) Oσνν′(k)


]∗

(E.45)

using formula (E.30) and the equality Oσνν′(Sk) = OS−1σ

S−1νS−1ν′(k) because of the commutationproperty of O:

〈ψσSk,ν |O|ψσSk′,ν′〉 = 〈ψS−1σk,S−1ν |S†OS|ψS−1σ

k′,S−1ν′〉= 〈ψS−1σ

k,S−1ν |S−1OS|ψS−1σk′,S−1ν′〉

= 〈ψS−1σk,S−1ν |O|ψS−1σ

k′,S−1ν′〉 since [O,S] = 0

(E.46)

Moreover, for each k point, the set to which belong the band indices ν is the following:

Ek = ν|Emin < εσkν < Emax (E.47)

Since εσSk,ν = εS−1σ

k,S−1ν , it implies that ESk = Ek and then the summation over ν is finally inde-pendent of the symmetry operation S.

• and similarly, when S ∈ ΘG:

[(Oα,σl

)mm′

]ΘG=

∑

S∈ΘG

∑

k∈IBZ

∑

ν,ν′



]∗

=∑

S∈ΘG

∑

n,n′

Dl(S)m,n[(

OS−1α,S−1σl

)nn′

]∗unsym

[Dl(S−1)n′,m′

]∗

(E.48)

using formula (E.38) and the following relation:

〈ψσSk,ν |O|ψσSk′,ν′〉 = 〈ψσSk,ν |OS|ψS−1σk′,S−1ν′〉

=[〈ψS−1σ

k,S−1ν |S−1OS|ψS−1σk′,S−1ν′〉

]∗since 〈Su|v〉 =

[〈u|S†v〉

]∗

=[〈ψS−1σ

k,S−1ν |O|ψS−1σk′,S−1ν′〉

]∗since [O,S] = 0

(E.49)

Moreover, we can also write (E.48) as follows:

[(Oα,σl

)mm′

]ΘG=

∑

R∈G

∑

n,n′

Dl(ΘR)m,n

[(OR−1α,−R−1σl

)nn′

]∗unsym

[Dl(R−1Θ−1)n′,m′

]∗

(E.50)


since the time reversal operator Θ has no effect on the atomic position α and reverses the spinΘσ = −σ. However, the compound is assumed to be paramagnetic and thus the spin inversioncan be ommited. As a result,

[(Oα,σl

)mm′

]ΘG=

∑

n,n′

Dl(Θ)m,n[(Oα,σl

)nn′

]G∗ [Dl(Θ−1)n′,m′

]∗(E.51)

where the time-reversal operator Θ and the previous term[(Oα,σl

)mm′

]G appear now explicitly.

Finally, the term(Oα,σl

)mm′ of (E.43) can be calculated by the following expression:

(Oα,σl

)mm′ =

1

2

[(Oα

l )σσmm′ ]

G +∑

n,n′

Dl(Θ)m,n[(Oα,σl

)nn′

]G∗ [Dl(Θ−1)n′,m′

]∗

with[(Oα,σl

)mm′

]G=∑

S∈G

∑

n,n′

Dl(S)m,n[(

OS−1α,S−1σl

)nn′

]unsym

Dl(S−1)n′,m′

and[(Oα,σl

)mm′

]unsym

=∑

k∈IBZ

∑

ν,ν′



]∗

(E.52)

when the spin-orbit coupling is taken into account

Taking into account the spin-orbit coupling does not modify the “grey” magnetic point group – G+ΘG –of the compound. Nevertheless, it is now necessary to treat explicitly the spin indices:

(Oαl )σσ′


∑

k∈1BZ

∑

ν,ν′

∑

i,j=+,−

|ψikν〉Oijνν′(k)〈ψ

jkν′ |

|wασ′

lm′ 〉

=∑

k∈1BZ

∑

ν,ν′

∑

i,j

[Pα,σlm,ν(k)

]iOijνν′(k)


]j ∗(E.53)

where 1BZ is still the first Brillouin zone of the original point group G.

(Oαl )σσ′

mm′ =1

2

∑

S∈G

∑

k∈IBZ

∑

ν,ν′

∑

i,j

[Pα,σlm,ν(Sk)

]iOijνν′(Sk)

[Pα,σ

′

lm′,ν′(Sk)]j ∗

+1

2

∑

S∈ΘG

∑

k∈IBZ

∑

ν,ν′

∑

i,j

[Pα,σlm,ν(Sk)

]iOijνν′(Sk)

[Pα,σ

′


(E.54)

where IBZ denotes the irreducible Brillouin zone of the original point group G.

• when S ∈ G, one gets:[(Oα

l )σσ′

mm′

]G=

∑

S∈G

∑

n,n′

∑

τ,τ ′

Dl(S)m,nD12 (S)σ,τ

[(OS−1αl

)ττ ′nn′

]

unsym

Dl(S−1)n′,m′D 12 (S−1)τ ′,σ′

with[(Oα

l )σσ′

mm′

]unsym

=∑

k∈IBZ

∑

ν,ν′

∑

i,j

[Pα,σlm,ν(k)

]iOijνν′(k)

[Pα,σ

′

lm′,ν′(k)]j ∗

(E.55)

by using the formula (E.33) and the following equality:

〈ψiSk,ν |O|ψjSk′,ν′〉 =∑

a,b

D 12 (S)i,a〈ψak,S−1ν |O|ψbk′,S−1ν′〉D

12 (S−1)b,j (E.56)


• and when S ∈ ΘG, one gets:

[(Oα

l )σσ′

mm′

]ΘG=

∑

S∈ΘG

∑

n,n′

∑

ττ ′


[(OS−1αl

)ττ ′nn′

]∗

unsym

[Dl(S−1)n′,m′D 1

2 (S−1)τ ′,σ′

]∗

(E.57)

by using formula (E.40) and the relation:

〈ψiSk,ν |O|ψjSk′,ν′〉 =∑

a,b

D 12 (S)i,a

[〈ψak,S−1ν |O|ψbk′,S−1ν′〉

]∗ [D 1

2 (S−1)b,j

]∗(E.58)

The term (Oαl )σσ′

mm′ of (E.53) can finally be written as follows:

(Oαl )σσ′

mm′ =1

2

[(Oα

l )σσ′

mm′

]G+∑

n,n′

∑

τ,τ ′

Dl(Θ)m,nD12 (Θ)σ,τ

[(Oα

l )ττ ′

nn′

]G∗ [Dl(Θ−1)n′,m′D 1

2 (Θ−1)τ ′,σ′

]∗

(E.59)

since D 12 (Θ) = −iσy and D 1

2 (Θ−1) = D 12 (Θ†) = D 1

2 (Θ)T = −D 12 (Θ), this leads to:

(Oαl )

↑↑mm′ =

1

2

[(Oα

l )↑↑mm′

]G+∑

n,n′

Dl(Θ)m,n

[(Oα

l )↓↓nn′

]G∗ [Dl(Θ−1)n′,m′

]∗

(Oαl )

↓↓mm′ =

1

2

[(Oα

l )↓↓mm′

]G+∑

n,n′

Dl(Θ)m,n

[(Oα

l )↑↑nn′

]G∗ [Dl(Θ−1)n′,m′

]∗

(Oαl )

↑↓mm′ =

1

2

[(Oα

l )↑↓mm′

]G−∑

n,n′

Dl(Θ)m,n

[(Oα

l )↓↑nn′

]G∗ [Dl(Θ−1)n′,m′

]∗

(E.60)

E.4.2 Case of a spin-polarized calculation

without spin-orbit coupling

We consider now a spin-polarized calculation – with a “black and white” (type III) magnetic pointgroup H+Θ(G rH) –. In this case, we have to perform the following computation:

∀ σ ∈ ↑, ↓(Oα,σl

)mm′ = 〈wασlm |

∑

k∈1BZ

∑

ν,ν′

|ψσkν〉Oσνν′(k)〈ψσkν′ |

|wασlm′〉

=∑

k∈1BZ

∑

ν,ν′



]∗. δσσ′

=∑

S∈H

∑

k∈IBZ

∑

ν,ν′



]∗

+∑

S∈Θ(GrH)

∑

k∈IBZ

∑

ν,ν′



]∗

(E.61)


where 1BZ (IBZ ) is the first (respectively the irreducible) Brillouin zone of the original point groupG. Using the previous expressions (E.45) and (E.48), one obtains:

(Oα,σl

)mm′ =

∑

S∈H

∑

n,n′

Dl(S)m,n[(

OS−1α,S−1σl

)nn′

]unsym

Dl(S−1)n′,m′

+∑

S∈Θ(GrH)

∑

n,n′

Dl(S)m,n[(

OS−1α,S−1σl

)nn′

]∗unsym

[Dl(S−1)n′,m′

]∗

with[(Oα,σl

)mm′

]unsym

=∑

k∈IBZ

∑

ν,ν′



]∗

(E.62)

If we assume that the magnetization axis of the compound lies along the z-axis, the halving subgroupH of G is defined as the symmetry operations which do not change the orientation of the magnetizationand thus keep the z-axis invariant. As a result,

S ∈ H ⇐⇒ βS = 0 and R ∈ G rH ⇐⇒ βR = π⇐⇒ ∀ σ ∈ ↑, ↓ Rσ = σ ⇐⇒ ∀ σ ∈ ↑, ↓ Sσ = −σ

where βS is the Euler angle β of the symmetry operation S. Since Θσ = −σ too, we finally can write:

∀ σ ∈ ↑, ↓(Oα,σl

)mm′ =

∑

S∈H

∑

n,n′

Dl(S)m,n[(

OS−1α,σl

)nn′

]unsym

Dl(S−1)n′,m′

+∑

R∈GrH

∑

n,n′

Dl(ΘR)m,n

[(OR−1α,σl

)nn′

]∗unsym

[Dl(R−1Θ−1)n′,m′

]∗

(E.63)

when the spin-orbit coupling is taken into account

AS previously, taking into account the spin-orbit coupling does not modify the “black and white”magnetic point group of the system but the spin indices must now be explicitly treated.

(Oαl )σσ′


∑

k∈1BZ

∑

ν,ν′

∑

i,j=+,−

|ψikν〉Oijνν′(k)〈ψ

jkν′ |

|wασ′

lm′ 〉

=∑

k∈1BZ

∑

ν,ν′

∑

i,j

[Pα,σlm,ν(k)

]iOijνν′(k)


]j ∗

=∑

S∈G

∑

k∈IBZ

∑

ν,ν′

∑

i,j

[Pα,σlm,ν(Sk)

]iOijνν′(Sk)

[Pα,σ

′


+∑

S∈Θ(GrH)

∑

k∈IBZ

∑

ν,ν′

∑

i,j

[Pα,σlm,ν(Sk)

]iOijνν′(Sk)

[Pα,σ

′


(E.64)


with 1BZ (IBZ )the first (respectively the irreducible) Brillouin zone of the original point group G.Using the previous expressions (E.55) and (E.57), one obtains:

(Oαl )σσ′

mm′ =∑

S∈G

∑

n,n′

∑

τ,τ ′


[(OS−1αl

)ττ ′nn′

]

unsym

Dl(S−1)n′,m′D 12 (S−1)τ ′,σ′

+∑

S∈Θ(GrH)

∑

n,n′

∑

ττ ′


[(OS−1αl

)ττ ′nn′

]∗

unsym

[Dl(S−1)n′,m′D 1

2 (S−1)τ ′,σ′

]∗

with[(Oα

l )σσ′

mm′

]unsym

=∑

k∈IBZ

∑

ν,ν′

∑

i,j

[Pα,σlm,ν(k)

]iOijνν′(k)

[Pα,σ

′

lm′,ν′(k)]j ∗

(E.65)

This expression can be simplified further if we assume that magnetization axis of the compoundlies along the z-axis. In this case, when S belongs to the subgroup H (β = 0), we indeed have in thebasis | ↑〉, | ↓〉:

D 12 (S) =

(cos β2 e


2 eiα−γ

2

sin β2 e


iα+γ2

)=

(e−i

α+γ2 0

0 eiα+γ2

)(E.66)

whereas when R belongs to the subgroup G rH (β = π),

D 12 (R) =

(cos β2 e


2 eiα−γ

2

sin β2 e


iα+γ2

)=

(0 −eiα−γ

2

e−iα−γ2 0

)(E.67)

thus leading to:

D 12 (S) = D 1

2 (ΘR) =

(0 −11 0

)(0 −eiα−γ

2

e−iα−γ2 0

)=

(−eiα−γ

2 0

0 −e−iα−γ2

)(E.68)

As a result, we can rewrite (E.65) as follows3:

(Oαl )

↑↑mm′ =

∑

S∈G

∑

n,n′

Dl(S)m,n[(

OS−1αl

)↑↑nn′

]

unsym

Dl(S−1)n′,m′

+∑

R∈(GrH)

∑

n,n′

Dl(ΘR)m,n

[(OR−1αl

)↑↑nn′

]∗

unsym

[Dl(R−1Θ−1)n′,m′

]∗

(Oαl )

↓↓mm′ =

∑

S∈G

∑

n,n′

Dl(S)m,n[(

OS−1αl

)↓↓nn′

]

unsym

Dl(S−1)n′,m′

+∑

R∈(GrH)

∑

n,n′

Dl(ΘR)m,n

[(OR−1αl

)↓↓nn′

]∗

unsym

[Dl(R−1Θ−1)n′,m′

]∗

(Oαl )

↑↓mm′ =

∑

S∈G

∑

n,n′

Dl(S)m,n[(

OS−1αl

)↑↓nn′

]

unsym

Dl(S−1)n′,m′ e−i(α+γ)

+∑

R∈(GrH)

∑

n,n′

Dl(ΘR)m,n

[(OR−1αl

)↑↓nn′

]∗

unsym

[Dl(R−1Θ−1)n′,m′

]∗ei(α−γ)

(E.69)

3We remind that if S belongs to the subgroup Θ(G rH), D 1

2 (S−1) = D 1

2 (S†) = D 1

2 (S)T (= D 1

2 (S) here).

Appendix F

From many-body spin-orbit interactionsto one-electron spin-orbit coupling

The “non-relativistic limit” of the Dirac equation up to the order (v/c)2 – and the “scalar relativisticapproximation” – was introduced in section 3.3. In standard treatments of the spin-orbit couplingwithin the DFT framework, equations (3.43) and (3.45) are used with the effective Kohn-Sham poten-tial VKS(r) instead of the mere external potential V (r). As a result, the many-body effects induced bythe Coulomb repulsion are included in this approach via the Hartree potential VH(r) and the exchange-correlation potential V xc(r). In this appendix, we address the question of how this spin-orbit couplingterm – with VKS(r) instead of V (r) – can be considered as a mean-field approximation to many-bodyinteractions.

Indeed, if one considers the non-relativistic limit of the Dirac-Coulomb-Breit Hamiltonian [26],other many-body terms appear in addition to the “standard” electron-electron Coulomb interaction.Among them, the “spin-same-orbit” and “spin-other-orbit” terms are the general many-body expressionof the spin-orbit interaction. The standard one-body term HSO term can thus be seen as a mean-fieldapproximation of the complete “spin-orbit interactions” which arise in the many-body problem.

In this appendix, we first introduce the many-body spin-orbit interaction terms, called “spin-same-orbit” and “spin-other-orbit”. They must be taken into account in a scalar relativistic approximationof the Dirac-Coulomb-Breit Hamiltonian [26]. With the help of a generalization of Hedin’s equationformalism [15, 65], we derive the generalized Hartree and Fock terms associated to these interactionsand calculated the screened interaction. These developments are the first step towards a full treatmentof “many-body spin-orbit interactions”, which is however beyond the frame of this thesis.

F.1 The spin-same-orbit and spin-other-orbit interactions

F.1.1 Introduction of these interaction terms

According to Slater [153] and Breit [26], taking into account the magnetic effects in the many-bodyproblem with a description of the electrons by the Dirac equation makes the following terms appear1:

the Coulomb interaction1

2

∑

i,j

e2

4πε0

1

rij, (F.1)

1with the definition s = 12σ, where σi (i=x,y,z) are the Pauli matrices.

161

162 APPENDIX F. MANY-BODY SPIN-ORBIT INTERACTIONS

the orbit-orbit interaction

1

2

∑

i,j

e2~2

4πε0m20c

2

[− (−i∇i) · (−i∇j)

2rij(F.2)

− [(ri − rj) · (−i∇i)][(ri − rj) · (−i∇j)]

2r3ij+

1

4

(ri − rj) · ∇i + (rj − ri) · ∇j

r3ij

],

the spin-same-orbit interaction

1

2

∑

i,j

− e2~2

4πε0m20c

2

si · [(ri − rj)× (−i∇i)] + sj · [(rj − ri)× (−i∇j)]

2r3ij, (F.3)

the spin-other-orbit interaction

1

2

∑

i,j

e2~2

4πε0m20c

2

si · [(ri − rj)× (−i∇j)] + sj · [(rj − ri)× (−i∇i)]

r3ij, (F.4)

the spin-spin interaction

1

2

∑

i,j

e2~2

4πε0m20c

2

[si · sjr3ij

− 3[si · (ri − rj)][sj · (rj − ri)]

r5ij

]. (F.5)

However, the spin-orbit effects are generally supposed to be greater than those due to the other terms,except of course the Coulomb interaction. That is why we will only consider the spin-same-orbit andthe spin-other orbit contributions in the following. The interaction term of our system is then:

V =1

2

∑

i,j

e2

4πε0

1

rij+

e2~

8πε0m20c

2[∇ri

(1

rij

)× (pi − 2pj)] · σi

=1

2

∑

i,j

e2

4πε0

1

rij+

e2~

8πε0m20c

2[∇ri

(1

rij

)× pi] · [σi + 2σj ]. (F.6)

By performing an adimensionalization with respect to εF for the energies and kF for the wave numbers,we find the following expressions where the relation between kF and the Bohr radius a0 is still to define:

vC =2

a20k2F

1

rεF and vSO = − 2α2

a0kF

1

r3[r × p] · 1

2σ εF . (F.7)

In the Hartree atomic units, the conventions lead to:

εF = (~2k2F )/(2m) = mc2α2 i.e. k2Fa20 = 2. (F.8)

In the Rydberg atomic units however, the fundamental relation is:

εF = (~2k2F )/(2m) = (1/2)mc2α2 i.e. k2Fa20 = 1. (F.9)

We will use the Hartree convention in the following and will then replace (e2~2)/(4πε0m20c

2) by α2√2.

F.1. THE SPIN-SAME-ORBIT AND SPIN-OTHER-ORBIT INTERACTIONS 163

F.1.2 Pauli matrices representation of a general interaction

An interaction is a two-body operator which has the following general form in second quantization:

V =1

2

∑

α,β,γ,η

∫∫ ∫∫d1d2d3d4 vαβγη(1, 2, 3, 4)ψ

†α(1)ψ

†β(2)ψη(4)ψγ(3) with i = (~ri, ti) (F.10)

with the two essential properties:

• vαβγη(1, 2, 3, 4) must be proportional to δ(t3 − t1)δ(t4 − t2) because of the conservation of thenumber of particles at each time. On the contrary, a term δ(t2−t1) appears only if the interactionis assumed instantaneous.In order to keep in mind this dependence on only two time parameters, we will use the followingshortcuts [i, j] = (ri, rj , ti) if ti = tj so that we can write:

vαβγη(1, 2, 3, 4) = vαβγη([1, 3]; [2, 4])δ(t3 − t1)δ(t4 − t2) (F.11)

• vαβγη(1, 2, 3, 4) = vβαηγ(2, 1, 4, 3) or vαβγη([1, 3]; [2, 4]) = vβαηγ([2, 4]; [1, 3]) because of the in-variance under particle interchange.

This interaction can be expanded in the Pauli matrices (σX ,σY ,σZ) and unit matrix (σ0 = Id) as:

vαβγη([1, 3]; [2, 4]) =∑

I,J=0,X,Y,Z

σIαγ vIJ([1, 3]; [2, 4]) σJβη (F.12)

In particular, the interaction term we consider can be written in this form:

vCαβγη([1, 3]; [2, 4]) = σ0αγδ(r3 − r1)δ(t2 − t1)

r12δ(r4 − r2)σ

0βη (F.13)

vSSOαβγη([1, 3]; [2, 4]) = σ0αγiα2

2√2[∇r2

(δ(t2 − t1)

r12

)×∇r4δ(r4 − r2)]Iδ(r3 − r1) σIβη

+ σIαγiα2

2√2[∇r1

(δ(t2 − t1)

r12

)×∇r3δ(r3 − r1)]Iδ(r4 − r2) σ0βη (F.14)

vSOOαβγη([1, 3]; [2, 4]) = − σ0αγiα2

2√2[∇r2

(δ(t2 − t1)

r12

)×∇r3δ(r3 − r1)]Iδ(r4 − r2) 2σ

Iβη

−2σIαγiα2

2√2[∇r1

(δ(t2 − t1)

r12

)×∇r4δ(r4 − r2)]Iδ(r3 − r1) σ0βη (F.15)

or, to be more clear:

vC00([1, 3]; [2, 4]) = δ(r3 − r1)δ(t2 − t1)

r12δ(r4 − r2)

(F.16)

vSSO0J ([1, 3]; [2, 4]) =iα2

2√2[∇r2

(δ(t2 − t1)

r12

)×∇r4δ(r4 − r2)]Iδ(r3 − r1) ∀J ∈ X,Y, Z

vSSOI0 ([1, 3]; [2, 4]) =iα2

2√2[∇r1

(δ(t2 − t1)

r12

)×∇r3δ(r3 − r1)]Iδ(r4 − r2) ∀I ∈ X,Y, Z

(F.17)

vSOO0J ([1, 3]; [2, 4]) = −2iα2

2√2[∇r2

(δ(t2 − t1)

r12

)×∇r3δ(r3 − r1)]Iδ(r4 − r2) ∀J ∈ X,Y, Z

vSOOI0 ([1, 3]; [2, 4]) = −2iα2

2√2[∇r1

(δ(t2 − t1)

r12

)×∇r4δ(r4 − r2)]Iδ(r3 − r1) ∀I ∈ X,Y, Z


We draw the reader’s attention to the gradient operator in the expression ∇ri(δ(ti−tj)rij

): this is justa shorter notation where the operator acts only on the following bracket contents. On the contrary,∇rjδ(j − i) is an operator which acts on the whole expression, including creation and annihilationoperators. Moreover, it is useful to notice that:

vSSO0I ([1, 3]; [2, 4]) = vSSOI0 ([2, 4]; [1, 3]) and vSOO0I ([1, 3]; [2, 4]) = vSOOI0 ([2, 4]; [1, 3]). (F.18)

This equality ensures the invariance under particle exchange of the interactions.

F.2 Spin-Hedin equations in their most general form

F.2.1 A brief history on Hedin’s equations

Inspired by the previous works of Schwinger, Hedin derived a closed set of equations for the electronicGreen’s function and self-energy, the screened Coulomb interaction and the polarization of a solid in1965 [65]:

Σ(1, 2) = i~

∫∫d3d4 G(1, 3+)W (1, 4)Λ(3, 2, 4)

G(1, 2) = G0(1, 2) +

∫∫d3d4 G0(1, 3)Σ(3, 4)G(4, 2)

P (1, 2) = −i~∫∫

d3d4 G(1, 3)Λ(3, 4, 2)G(4, 1+) (F.19)

W (1, 2) = v(1, 2) +

∫∫d3d4 v(1, 3)P (3, 4)W (4, 2)

Λ(1, 2, 3) = δ(1− 2)δ(2− 3) +

∫∫ ∫∫d4d5d6d7

δΣ(1, 2)

δG(4, 5)G(4, 6)G(7, 5)Λ(6, 7, 3)

with the convention i = (~ri, ti, σi). In these equations, Σ(1, 2) is the self energy, P (1, 2) the polarization,W (1, 2) the screened interaction and Λ(1, 2, 3) the vertex function. These equations are exact but themost common and simplest use of them is made by the so-called “GW approximation” which consistsin writing:

Σ(1, 2) = i~ G(1, 2)W (2, 1)

G(1, 2) = G0(1, 2) +

∫∫d3d4 G0(1, 3)Σ(3, 4)G(4, 2)

P (1, 2) = −i~ G(1, 2)G(2, 1+) (F.20)

W (1, 2) = v(1, 2) +

∫∫d3d4 v(1, 3)P (3, 4)W (4, 2)

Λ(1, 2, 3) = δ(1− 2)δ(2− 3).

In 2008, a generalization of equations (F.19) for describing systems containing spin interactions,

F.2. SPIN-HEDIN EQUATIONS IN THEIR MOST GENERAL FORM 165

such as spin-orbit and spin-spin interactions, was developed by Aryasetiawan and Biermann [15]:

Σαβ(1, 2) = −σIαηGηγ(1, 3)ΛJγβ(3, 2, 4)WJI(4, 1)

Gαβ(1, 2) = G0αβ(1, 2) +G0

αγ(1, 3)Σγη(3, 4)Gηβ(4, 2)

PIJ(1, 2) = σIαβGβγ(1, 3)ΛJγη(3, 4, 2)Gηα(4, 1

+) (F.21)

WIJ(1, 2) = vIJ(1, 2) + vIK(1, 3)PKL(3, 4)WLJ(4, 2)

ΛIαβ(1, 2, 3) = δ(1− 2)δ(2− 3)σIαβ +δΣαβ(1, 2)

δGγη(4, 5)Gγκ(4, 6)Λ

Iκµ(6, 7, 3)Gµη(7, 5).

where i = (~ri, τi), τ is the imaginary time and σI are the Pauli spin matrices. Repeated indices aresummed and repeated variables are integrated.

F.2.2 Spin-Hedin’s equations for a general interaction term

Let’s now rewrite the spin-Hedin’s equations with the general expression (F.12) for the interactionterm. To begin, the Heisenberg equations of motion are:

−i~∂tψκ(5) = [H, ψκ(5)] with H = H0 + V

= −h0κβ(5, 1)ψβ(1)−1

2

(vακγη(1, 5, 3, 4)− vκαγη(5, 1, 3, 4)

)ψ†α(1)ψη(4)ψγ(3)

= −h0κβ(5, 1)ψβ(1)−1

2

(vακγη(1, 5, 3, 4)− vακηγ(1, 5, 4, 3)

)ψ†α(1)ψη(4)ψγ(3)

= −h0κβ(5, 1)ψβ(1)− vακγη(1, 5, 3, 4)ψ†α(1)ψη(4)ψγ(3)

= −h0κβ(5, 1)ψβ(1)− vακγη([1, 3]; [5, 4])[ψ†α(1)ψη(4)ψγ(3)

]t3=t1t4=t5

(F.22)

where we have used the invariance under particle interchange. As a result, one gets:

−i~∂tGκµ(5, 6) = −i~∂t(−i~< T [ψκ(5)ψ

†µ(6)] >

)

= −h0κβ(5, 1)Gβµ(1, 6) + i~ vακγη([1, 3]; [5, 4])G(2)ηµγα(4, 6, 3, 1

+)− δ(5− 6)δκµ

(F.23)

where, as usual, G(2)αβγη(1, 2, 3, 4) = (−i/~)2 < T [ψα(1)ψη(3)ψ

†γ(4)ψ

†β(2)] > and the notation 1+ means

t1 = t3 + 0+.2

We will then use the Schwinger functional derivative technique, with the probing field:

φ =

∫∫d1d2 ψ†

α(1)(ϕI([1, 2]).δ(t2 − t1)

)σIαβ ψβ(2) in S = T

[exp

( −i~φ)]. (F.24)


δGαβ(1, 2)

δϕI([3, 4])=

[Gαβ(1, 2)Gηγ(3, 4

+)−G(2)αβηγ(1, 2, 4, 3

+)]σIγη (F.25)

andδGαβ(1, 2)

δϕI([3, 4])= −

∫∫d5d6 Gαγ(1, 5)

δG−1γη (5, 6)

δϕI([3, 4])Gηβ(6, 2). (F.26)

2We remind the reader that the studied interactions contains a term δ(t1 − t3).


Consequently, the mass operator can be defined as:

Mκη(5, 4)Gηµ(4, 6) = −i~ vακγη([1, 3]; [5, 4])G(2)ηµγα(4, 6, 3, 1

+)

= −i~ σIαγ vIJ([1, 3]; [5, 4]) σJκηG(2)ηµγα(4, 6, 3, 1

+)

= V HJ (5, 4)σJκηGηµ(4, 6) + i~ vIJ([1, 3]; [5, 4])σ

Jκη

δGηµ(4, 6)

δϕI([1, 3])

=

[V HJ (5, 4)σJκη +Σκη(5, 4)

]Gηµ(4, 6) (F.27)

with the following exact expression for the self-energy:

Σαβ(1, 2) = −i~ vIJ([0, 3], [1, 4])σJαγGγη(4, 5)δG−1

ηβ (5, 2)

δϕI([0, 3])(F.28)

and the generalized Hartree potential V HI (1, 3) = V H

I ([1, 3]).δ(t3 − t1)

with V HI ([1, 3]) = −i~ vJI([2, 4]; [1, 3])Gηγ(2, 4+)σJγη

= −i~ vJI([2, 4]; [1, 3])GJ(2, 4+) = vIJ([1, 3]; [2, 4])ρJ(2, 4+) (F.29)

where ρJ(2, 4+) = −i~ GJ(2, 4+) is actually[ρJ(x4,x2)

]t2

, i.e. the J-component of the density matrixin the Pauli matrices formalism. In particular, when x4 = x2, ρ0(2, 2+) =

[ρ0(x2)

]t2

is the totalcharge density in (x2, t2), and ρI(2, 2+) =

[ρI(x2)

]t2

(I=X,Y,Z) is the density of spin along the I-axisin (x2, t2).

We define now

• the total field as: ΦI([1, 2]) = ϕI([1, 2]) + V HI ([1, 2])

• the vertices3

ΛIαβ(1, 2, [3, 4]) = −δG−1

αβ(1, 2)

δΦI([3, 4])= −

δ[G0]−1αβ(1, 2)

δΦI([3, 4])+δΣαβ(1, 2)

δΦI([3, 4])

=[δ(3− 1)δ(4− 2)σIαβ

]t3=t4

+δΣαβ(1, 2)

δΦI([3, 4])(F.30)

• the dielectric function

ε−1IJ ([1, 2]; [3, 4]) =

δΦI([1, 2])

δϕJ([3, 4])=[δ(3− 1)δ(4− 2)δIJ

]t1=t2t3=t4

+δV H

I ([1, 2])

δϕJ([3, 4])(F.31)

• the screened interaction as: WIJ([1, 3]; [2, 4]) = ε−1IK([1, 3]; [5, 6])vKJ([5, 6]; [2, 4])

• and the polarization

PIJ([1, 2]; [3, 4]) =δρI(2, 1+)

δΦJ([3, 4])= −i~ σIαβ

δGβα(2, 1+)

δΦJ([3, 4])(F.32)

3We remind the reader that:

[G0]−1αβ(1, 2) = H0

αβ(1, 2) = i~∂t − h0αβ(1, 2)−

[V HI ([1, 2])δ(t2 − t1)

]σIαβ −

[ϕI([1, 2])δ(t2 − t1)

]σIαβ

= i~∂t − h0αβ(1, 2)−

[ΦI([1, 2])δ(t2 − t1)

]σIαβ

F.3. GENERALIZED HARTREE POTENTIAL FOR OUR SYSTEM 167

which enable us to write the generalized spin-Hedin’s equations4:

Σαβ(1, 2) = i~ σIαη

[Gηγ(4, 3)

]t4=t1

ΛJγβ(3, 2, [5, 6])WJI([5, 6]; [1, 4])

Gαβ(1, 2) = G0αβ(1, 2) +G0

αγ(1, 3)Σγη(3, 4)Gηβ(4, 2)

PIJ([1, 3]; [2, 4]) = −i~ σIαβ[Gβγ(3, 5)

]t3=t1

ΛJγη(5, 6, [2, 4])Gηα(6, 1+) (F.33)

WIJ([1, 3]; [2, 4]) = vIJ([1, 3]; [2, 4]) + vIK([1, 3]; [5, 6])PKL([5, 6]; [7, 8])WLJ([7, 8]; [2, 4])

ΛIαβ(1, 2, [3, 4]) =[δ(3− 1)δ(4− 2)σIαβ

]t4=t3

+δΣαβ(1, 2)

δGγη(5, 6)Gγκ(6, 7)Λ

Iκµ(7, 8, [3, 4])Gµη(8, 6).

As a consequence, the GW approximation becomes:

ΛIαβ(1, 2, [3, 4]) =[δ(3− 1)δ(4− 2)σIαβ

]t4=t3

PIJ([1, 3]; [2, 4]) = −i~ σIαβ[Gβγ(3, 2)

]t3=t1

σJγη

[Gηα(4, 1

+)]t4=t2

(F.34)

Σαβ(1, 2) = i~ σIαη

[Gηγ(4, 3)

]t4=t1t3=t2

σJγβWJI([3, 2]; [1, 4]).

F.3 Generalized Hartree potential for our system

Our first approach of the problem will be to understand the meaning of the mean-field potential termsthat we will find with the considered interactions.

F.3.1 The Coulomb interaction

for I = X,Y, Z V CHI ([1, 3]) = vCIJ([1, 3]; [2, 4])ρ

J(2, 4+) = 0 (F.35)

and V CH0 ([1; 3]) = vC0J([1, 3]; [2, 4])ρ

J(2, 4+) = vC00([1, 3]; [2, 4])ρ0(2, 4+)

= δ(r3 − r1)δ(t2 − t1)

r12δ(r4 − r2) . ρ

0(2, 4+)

=

∫d2

ρ0(2, 2+)

r12δ(t2 − t1) . δ(r3 − r1) (F.36)

where ρ0(2, 2+) = G↑↑(2, 2+) + G↓↓(2, 2

+). The result is thus the expected one: in a mean-fieldapproach, an electron feels the potential created by the mean charge distribution ρ0; the operator isdiagonal – proportional to δ(3− 1) – and can be rewritten in its usual form:

V CH0 (1) =

∫dr2

ρ0(2, 2+)

r12δ(t2 − t1) or V CH

0 (r1, t1) =

∫dr2

ρ0(r2, t1)

r12. (F.37)

4To get the equivalent equations in imaginary time, it’s enough to replace −i~ σIαβ by σI

αβ in the expression of thepolarization PIJ([1, 3]; [2, 4]) and the self-energy Σαβ(1, 2).


F.3.2 The spin-same-orbit interaction

for I = X,Y, Z V SSOHI ([1, 3]) = vSSOIJ ([1, 3]; [2, 4])ρJ(2, 4+) = vSSOI0 ([1, 3]; [2, 4])ρ0(2, 4+)

=iα2

2√2

[∇r1

(δ(t2 − t1)

r12

)×∇r3δ(r3 − r1)

]Iδ(r4 − r2) . ρ

0(2, 4+)

=iα2

2√2

[∇r1

(∫d2

ρ0(2, 2+)

r12δ(t2 − t1)

)×∇r3δ(r3 − r1)

]I

=iα2

2√2

[∇r1

(V CH0 (r1, t1)

)×∇r3δ(r3 − r1)

]I

(F.38)

or V SSOHI (r1, t1; r3, t3) =

α2

2√2

[∇r1

(V CH0 (r1, t1)

)×[i∇r3δ(r3 − r1)

]]I. δ(t3 − t1).

These three terms describe the spin-same-orbit interaction between an electron and the mean chargedistribution ρ0. They can be easier understood when we write:

∑

I=X,Y,Z

V SSOHI ([1, 3])σIαγ =

α2

2√2

[∇r1

(V CH0 (r1, t1)

)×[i∇r3δ(r3 − r1)

]]· σαγ

i.e. “ V SSOH(r1, t1) ” = − α2

~√2

[∇r1

(− V CH

0 (r1, t1))× p1

]· s1 with "less strict" notations.

We recognize the usual spin-orbit coupling between an electron and an external potential, which ishere replaced by the Hartree potential of the system. We remind the reader that the expression forthe spin-orbit coupling between an electron and an external potential (created by the nuclei of a solidfor instance) is the following5:

V SO(r, t) = − e~

2m20c

2

[∇r

(Vnuclei(r, t)

)× p

]· s = − e2~

8πε0m20c

2

[∇r

(∑

N

ZNδ(t− tN )

|r − RN |)× p

]· s

= − α2

~√2

[∇r

(∑

N

ZNδ(t− tN )

|r − RN |)× p

]· s. (F.39)

Thus the previous Hartree terms can be seen as taking into account the electrostatic screening in thespin-orbit coupling since:

V SO(r1, t1) + V SSOH(r1, t1) = − α2

~√2

[∇r1

(∑

N

ZNδ(t1 − tN )

|r1 − RN |− V CH

0 (r1, t1))× p1

]· s1. (F.40)

On the contrary, the last Hartree term V SSOH0 ([1, 3]) is diagonal in real space – i.e. it is a local

operator – and gives the interaction of the mean spin-density of the system with the electric field

5The spin-orbit coupling is a non-local one-body operator. It has the following form with the chosen formalism:

∑

I=X,Y,Z

V SOI ([1, 3])σI

αγ = − α2

2√2

[∇r1

(Vnuclei(r1, t1)

)×

[i∇r3

δ(r3 − r1)]]

· σαγ

V SO0 ([1, 3])σ0

αγ = 0

F.3. GENERALIZED HARTREE POTENTIAL FOR OUR SYSTEM 169

generated by an electron.

V SSOH0 ([1, 3]) = vSSO0J ([1, 3]; [2, 4])ρJ(2, 4+)

=iα2

2√2

[∇r2

(δ(t2 − t1)

r12

)×∇r4δ(r4 − r2)

]Jδ(r3 − r1) . ρ

J(2, 4+)

=α2

2√2

∫d2 ∇r2

(δ(t2 − t1)

r12

)·[∫

dr4 i∇r4δ(r4 − r2)× ~ρ(2, 4+)

]. δ(r3 − r1)

=α2

2√2

∫d2 ∇r2

(δ(t2 − t1)

r12

)·[−icurlr4 ~ρ(2, 4+)

]4=2

. δ(r3 − r1)

(F.41)

or V SSOH0 (r1, t1) =

α2

2√2

∫dr2 ∇r2

(1

r12

).[p2 × ~ρ(r2, t1)]

=α2

2√2

∫dr2 ∇r2

(1

r12

). [−icurlr2 ~ρ(r2, t1)] . (F.42)

The mean energy associated to these Hartree terms has the following property:

< V SSOH0 ([1, 3])σ0αγ > =

α2

2√2

∫dr2

[∇r2

(V CH0 (r2, t1)

)·[−icurlr4 ~ρ(2, 4+)

]4=2

= <∑

I=X,Y,Z

V SSOHI ([1, 3])σIαγ > . (F.43)

There is thus an equipartition of the energy between the two aspects of the Hartree terms induced bythe spin-same-orbit interaction.

F.3.3 The spin-other-orbit interaction

For I = X,Y, Z

V SOOHI ([1, 3]) = vSOOIJ ([1, 3]; [2, 4])ρJ(2, 4+) = vSOOI0 ([1, 3]; [2, 4])ρ0(2, 4+)

= −2iα2

2√2

[∇r1

(δ(t2 − t1)

r12

)×∇r4δ(r4 − r2)

]Iδ(r3 − r1) . ρ

0(2, 4+)

= −2α2

2√2

∫dr2

[∇r1

(δ(t2 − t1)

r12

)×(− i∇r4ρ

0(2, 4+))4=2

]I. δ(r3 − r1)

= −2α2

2√2

[∫dr2

[r1 − r2]× [−i∇r4ρ0(2, 4+)]4=2 & t2=t1

r312

]

I

. δ(r3 − r1)

(F.44)

or V SOOHI (r1, t1) =

e~

m0

[µ04π

∫dr2

[r1 − r2]× j(r2, t1)

r312

]

I

. δ(3− 1).

(F.45)

As previously, these three terms describe the spin-other-orbit interaction between an electron and themean charge distribution ρ0. However, the Biot-Savart law6 can be recognized, which enables us to

6The usual definition of the electrical current is

j(r, t) = − e~

2m0

[ψ†(r, t)

(− i∇ψ(r, t)

)−

(− i∇ψ†(r, t)

)ψ(r, t)

]


give a purely magnetic interpretation:

∑

I=X,Y,Z

V SOOHI (r1, t1)σ

Iαγ =

e~

2m0

[µ04π

∫dr2

[r1 − r2]× j(r2, t1)

r312

]· σαγ

= − gq

2m0B(r1, t1) ·

[12~σ]αγ

(F.46)

with the electron g-factor g = 2 and the electron charge q = −e. These terms thus describe the inter-action between the spin of an electron and the magnetic field induced by the mean charge currents inthe system.

On the contrary, the last Hartree term gives the interaction of the mean spin of the system withthe magnetic field generated by the "motion of one electron”.

V SSOH0 ([1, 3]) = vSSO0J ([1, 3]; [2, 4])ρJ(2, 4+)

= −2iα2

2√2

[∇r2

(δ(t2 − t1)

r12

)×∇r3δ(r3 − r1)

]Jδ(r4 − r2) . ρ

J(2, 4+)

= − e~

m0

[µ04π

∫dr2

r2 − r1r312

× ie~

2m0∇r3δ(r3 − r1)

]. ~ρ(r2, t1).

(F.47)

The mean energy associated to this term verifies also an equipartition of the energy between the twoaspects of the interaction:

< V SOOH0 ([1, 3])σ0αγ > =

∫dr2

e~

m0

[µ04π

∫dr1

[r2 − r1]× j(r1, t1)

r312

]. ~ρ(r2, t1)

= − gq

2m0

∫dr2 B(r2, t1) · S(r2, t1) with S(r2, t1) =

1

2~~ρ(r2, t1)

= <∑

I=X,Y,Z

V SOOHI (r1, t1)σ

Iαγ > . (F.48)

F.4 Exchange self-energy Σx – generalized Fock terms – in our system

In the usual Hedin’s formalism, the screened interaction W (1, 2) is split up into the bare Coulombpotential v(1, 2) and the induced potential Wc(1, 2). Using the GW approximation, the self-energy isthen written:

Σ(1, 2) = i~G(1, 2+)v(1, 2) + i~G(1, 2+)Wc(1, 2). (F.49)

The first term is referred to as the exchange (Fock) self-energy Σx(1, 2) and the second as the correlationself-energy Σc(1, 2). This section will now focus on the computation of the Fock terms we will obtainwith the considered interactions, in the generalized spin-Hedin’s formalism.

F.4.1 The Coulomb interaction

Since we use the same decomposition of the screened interaction WIJ([1, 3]; [2, 4]) and the GW approx-imation, the Fock self-energy Σxαβ(1, 2) has (of course) the same expression as usual:

ΣxC αβ(1, 2) = i~ σIαη [Gηγ(4, 3)]t4=t1t3=t2

σJγβvCJI([3, 2]; [1, 4])

= i~∑

ηγ

∫∫dr3dr4 σ0αη [Gηγ(4, 3)]t4=t1

t3=t2σ0γβδ(r2 − r3)

δ(t1 − t3)

r31δ(r4 − r1)

= i~δ(t1 − t2)

r12Gαβ(1, 2) (F.50)

F.4. GENERALIZED FOCK TERMS IN OUR SYSTEM 171

We remind the reader that the bare Coulomb interaction is instantaneous in the limit t2 = t+1 . We canthus rewrite Gαβ(1, 2) as Gαβ(1, 2+) in the previous expression.

ΣxC αβ(1, 2) = i~δ(t1 − t2)

r12Gαβ(1, 2

+) = −δ(t1 − t2)

r12< ψ†

β(2)ψα(1) >t2=t1 (F.51)

This argument will be also valid for the spin-same-orbit and spin-other-orbit interactions, which relyon the spatial gradient of the bare Coulomb interaction, and then will be used without repeating thisjustification.

F.4.2 The spin-same-orbit interaction

ΣxSSO αβ(1, 2) = i~ σIαη [Gηγ(4, 3)]t4=t1t3=t2

σJγβvSSOJI ([3, 2]; [1, 4])

= i~[σ0αη [Gηγ(4, 3)]t4=t1

t3=t2σJγβv

SSOJ0 ([3, 2]; [1, 4])

+σIαη [Gηγ(4, 3)]t4=t1t3=t2

σ0γβvSSO0I ([3, 2]; [1, 4])

]

= Σx (1)SSO αβ(1, 2) + Σ

x (2)SSO αβ(1, 2) (F.52)

In the following, we will compute separately these two terms in order to be clearer.

Σx (1)SSO αβ(1, 2) = i~ σ0αη [Gηγ(4, 3)]t4=t1

t3=t2σJγβ v

SSOJ0 ([3, 2]; [1, 4])

= − α2~

2√2

∑

γ

[∇r2

(δ(t1 − t2)

r12

)×∇r2Gαγ(1, 2)

]· σγβ

= − iα2

2√2

∑

γ

[∇r2

(δ(t1 − t2)

r12

)× < ∇r2

[ψ†γ(2)

]ψα(1) >t2=t1

]· σγβ (F.53)

Σx (2)SSO αβ(1, 2) = i~ σIαη [Gηγ(4, 3)]t4=t1

t3=t2σ0γβ v

SSO0I ([3, 2]; [1, 4])

= +α2~

2√2

∑

η

[∇r1

(δ(t1 − t2)

r12

)×∇r1Gηβ(1, 2)

]· σαη

= +iα2

2√2

∑

η

[∇r1

(δ(t1 − t2)

r12

)× < ψ†

β(2)∇r1

[ψη(1)

]>t2=t1

]· σαη (F.54)

These two computed terms have the following property:

Σx (1)SSO αβ(1, 2) = − iα2

2√2

∑

γ

[∇r2

(δ(t1 − t2)

r12

)× < ∇r2

[ψ†γ(2)

]ψα(1) >t2=t1

]· σγβ

=

[iα2

2√2

∑

γ

[∇r2

(δ(t1 − t2)

r12

)× < ψ†

α(1)∇r2

[ψγ(2)

]>t2=t1

]· σβγ

]∗

=[Σx (2)SSO βα(2, 1)

]∗=[Σx (2)SSO

]†αβ

(1, 2) (F.55)

As a consequence, the operator ΣxSSO = Σx (1)SSO +

[Σx (1)SSO

]† is hermitian and its mean value on a stateof the system is real.


F.4.3 The spin-other-orbit interaction

ΣxSOO αβ(1, 2) = i~ σIαη [Gηγ(4, 3)]t4=t1t3=t2

σJγβvSOOJI ([3, 2]; [1, 4])

= i~[σ0αη [Gηγ(4, 3)]t4=t1

t3=t2σJγβv

SOOJ0 ([3, 2]; [1, 4])

+σIαη [Gηγ(4, 3)]t4=t1t3=t2

σ0γβvSOO0I ([3, 2]; [1, 4])

]

= Σx (1)SOO αβ(1, 2) + Σ

x (2)SOO αβ(1, 2) (F.56)

As previously, we will compute separately these two contributions.

Σx (1)SOO αβ(1, 2) = i~ σ0αη [Gηγ(4, 3)]t4=t1

t3=t2σJγβ v

SOOJ0 ([3, 2]; [1, 4])

= −α2~√2

∑

γ

[∇r2

(δ(t1 − t2)

r12

)×∇r1Gαγ(1, 2)

]· σγβ

= − iα2

√2

∑

γ

[∇r2

(δ(t1 − t2)

r12

)× < ψ†

γ(2)∇r1

[ψα(1)

]>t2=t1

]· σγβ (F.57)

Σx (2)SOO αβ(1, 2) = i~ σIαη [Gηγ(4, 3)]t4=t1

t3=t2σ0γβ v

SOO0I ([3, 2]; [1, 4])

= +α2~√2

∑

η

[∇r1

(δ(t1 − t2)

r12

)×∇r2Gηβ(1, 2)

]· σαη

= +iα2

√2

∑

η

[∇r1

(δ(t1 − t2)

r12

)× < ∇r2

[ψ†β(2)

]ψη(1) >t2=t1

]· σαη (F.58)

With an analogous proof, it can be showed that these terms have the same property as highlightedin the previous section.

Σx (1)SOO αβ(1, 2) =

[Σx (2)SOO βα(2, 1)

]∗=[Σx (2)SOO

]†αβ

(1, 2) (F.59)

The operator ΣxSOO = Σx (1)SOO +

[Σx (1)SOO

]† is thus hermitian too and its mean value on a state of thesystem is real.

F.5 Expression of the screened interaction W

We now calculate the screened interaction W .

F.5.1 Matrix approach & Polarization computation

The expression of W in the generalized spin-Hedin’s equations can be understood as a generalizedmatrix product:

WIJ([1, 3]; [2, 4]) = vIJ([1, 3]; [2, 4]) + vIK([1, 3]; [5, 6])PKL([5, 6]; [7, 8])WLJ([7, 8]; [2, 4])

< 1, 3 I |W |2, 4 J > = < 1, 3 I |V |2, 4 J > + < 1, 3 I |I V |5, 6 K >< 5, 6 K |P |7, 8 L >< 7, 8 L|W |3, 4 J >

(F.60)

F.5. EXPRESSION OF THE SCREENED INTERACTION W 173

We can thus use the same expression as usual:

W = V + V P W =[1− V P

]−1V (F.61)

Using then this matrix-like notation, the operator V can be written as:

V =

v00 v0X v0Y v0ZvX0

vY 0 O

vZ0

with

v00 = vC00v0J = vSSO0J + vSOO0J ∀J ∈ X,Y, ZvI0 = vSSOI0 + vSOOI0 ∀I ∈ X,Y, Z

(F.62)

where each vIJ is actually a matrix (< 1, 3|vIJ |2, 4 >)(1,2,3,4). We will use the decomposition in thesame block matrices in the following.

In order to perform our computation further, we have to precise the form of the polarization matrixP too. In a first approach, we will use the GW approximation:

PIJ([1, 3]; [2, 4]) = −i~ σIαβ [Gβγ(3, 2)]t3=t1 σJγη

[Gηα(4, 1

+)]t4=t2

(F.63)

however, this expression strongly depends on the form of Gαβ(1, 2). For instance, considering a spin-diagonal Green function (but assuming that G↑↑ 6= G↓↓) leads to the following expressions:

G =

(G↑↑ O

O G↓↓

)then P =

P00 0 0 P0Z

0 PXX PXY 00 PY X PY Y 0

PZ0 0 0 PZZ

(F.64)

with

P00([1, 3]; [2, 4]) = −i~([G↑↑(3, 2)]t3=t1 [G↑↑(4, 1

+)]t4=t2 + [G↓↓(3, 2)]t3=t1 [G↓↓(4, 1+)]t4=t2

)

= PZZ([1, 3]; [2, 4])

P0Z([1, 3]; [2, 4]) = −i~([G↑↑(3, 2)]t3=t1 [G↑↑(4, 1

+)]t4=t2 − [G↓↓(3, 2)]t3=t1 [G↓↓(4, 1+)]t4=t2

)

= PZ0([1, 3]; [2, 4])

PXX([1, 3]; [2, 4]) = −i~([G↑↑(3, 2)]t3=t1 [G↓↓(4, 1

+)]t4=t2 + [G↓↓(3, 2)]t3=t1 [G↑↑(4, 1+)]t4=t2

)

= PY Y ([1, 3]; [2, 4])

PXY ([1, 3]; [2, 4]) = −i~(−i [G↑↑(3, 2)]t3=t1 [G↓↓(4, 1

+)]t4=t2 + i [G↓↓(3, 2)]t3=t1 [G↑↑(4, 1+)]t4=t2

)

= −PY X([1, 3]; [2, 4])

F.5.2 Computation of the screened interaction W

As the expression of the screened interaction is W =[1 − V P

]−1V = ε−1

V , the dielectric functionhas to be computed first.

ε = 1− V P =

(ε00 ε0JεI0 εIJ

)with ε00 = 1− v00P00 − v0ZPZ0 (F.65)

ε0X = −(v0XPXX + v0Y PY X) εX0 = −vX0P00

ε0Y = −(v0XPXY + v0Y PY Y ) εY 0 = −vY 0P00

ε0Z = −(v00P0Z + v0ZPZZ) εZ0 = −vZ0P00

εIJ =

1 0 −vX0P0Z

0 1 −vY 0P0Z

0 0 1− vZ0P0Z

As each term of the matrix ε is a matrix itself, the inversion operation must be done carefully, usingthe following identity:(A B

C D

)−1

=

(A−1 +A−1B∆−1CA−1 −A−1B∆−1

−∆−1CA−1 ∆−1

)with ∆ = D − CA−1B (F.66)


It appears then, taking into account that only terms of order O(α2) are relevant:

A−1ε =

[1− v00P00 − v0ZPZ0

]−1

≈[1− v00P00

]−1+[1− v00P00

]−1v0ZPZ0

[1− v00P00

]−1+O(α4)

(F.67)

and ∆−1ε ≈

1 0 vX0

[1− P00v00

]−1P0Z

0 1 vY 0

[1− P00v00

]−1P0Z

0 0 1 + vZ0[1− P00v00

]−1P0Z

As a result, the expression of the dielectric function is:

ε−1=[1− V P

]−1 ≈(ε−100 ε−1

0J

ε−1I0 ε−1

IJ

)+O(α4) (F.68)

with ε−100 ≈

[1− v00P00

]−1+[1− v00P00

]−1v0ZPZ0

[1− v00P00

]−1

+[1− v00P00

]−1v00P0ZvZ0P00

[1− v00P00

]−1

ε−10X ≈

[1− v00P00

]−1(v0XPXX + v0Y PY X)

ε−10Y ≈

[1− v00P00

]−1(v0XPXY + v0Y PY Y )

ε−10Z ≈

[1− v00P00

]−1(v00P0Z + v0ZPZZ) +

[1− v00P00

]−1v00P0ZvZ0

[1− P00v00

]−1P0Z

ε−1X0 ≈ vX0P00

[1− v00P00

]−1

ε−1Y 0 ≈ vY 0P00

[1− v00P00

]−1

ε−1Z0 ≈ vZ0P00

[1− v00P00

]−1

ε−1IJ = ∆−1

ε (I, J) ∀I, J ∈ X,Y, Z

and the expression of the screened interaction has the following form:

W = ε−1V ≈

(W00 W0J

WI0 O

)+O(α4) (F.69)

with W00 =[1− v00P00

]−1v00 +

[1− v00P00

]−1v0ZPZ0

[1− v00P00

]−1v00

+[1− v00P00

]−1v00P0ZvZ0

[1− P00v00

]−1

W0X =[1− v00P00

]−1v0X WX0 = vX0

[1− P00v00

]−1

W0Y =[1− v00P00

]−1v0Y and WY 0 = vY 0

[1− P00v00

]−1

W0Z =[1− v00P00

]−1v0Z WZ0 = vZ0

[1− P00v00

]−1

(F.70)

The conclusion of this last study is thus the appearance of a screening of the spin-same-orbit andspin-other-orbit interactions by the Coulomb interaction and the existence of two new terms in thecharge-charge channel, obtained by the coupling of the Coulomb and the spin-same-orbit and spin-other-orbit interactions.

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Couplage Spin-Orbite et Interaction de Coulombdans l’Iridate de Strontium Sr2IrO4

Thèse de Doctorat en Physique des Matériaux et des Milieux Denses Cyril Martins

Résumé

Cette thèse s’intéresse à l’interaction entre le couplage spin-orbite et les corrélations électroniques dans la matière

condensée. En effet, de plus en plus de matériaux – tels que les isolants topologiques ou les oxydes de métaux de

transition 5d à base d’iridium – présentent des propriétés pour lesquels l’interaction spin-orbite joue un rôle essentiel.

Parmi eux, l’iridate de strontium (Sr2IrO4) a récemment été décrit comme un “isolant de Mott régi par les effets spin-

orbite”: dans cette image, l’interaction de Coulomb entre les électrons et le couplage spin-orbite (ζSO ≈ 0.4 eV) se

combinent pour rendre le composé isolant.

Nous avons étudié la phase isolante paramagnétique de ce matériau avec l’approche LDA+DMFT, une méthode

qui combine la théorie de la fonctionnelle de la densité dans l’approximation de la densité locale (LDA) avec la théorie

du champ moyen dynamique (DMFT). Sr2IrO4 s’est avéré être un isolant de Mott pour une valeur raisonnable des

corrélations électroniques (U = 1, 4 eV) une fois que le couplage spin-orbite et les distorsions structurales du cristal ont

été pris en compte. En outre, nos résultats mettent en évidence les rôles respectifs joués par ces deux éléments dans

l’obtention d’un état isolant et montrent que seule leur action conjointe permet d’ouvrir un gap de Mott dans un tel

composé.

Afin de réaliser cette étude, le couplage spin-orbite a dû être inclus au sein du formalisme LDA+DMFT – plus

précisément, dans la définition des orbitales de Wannier sur lequel le problème local d’impureté repose. L’intérêt d’un tel

développement technique dépasse le cas de Sr2IrO4, cette implémentation, dite “LDA+SO+DMFT ”, pouvant être aussi

utilisée pour prendre en compte les corrélations électroniques dans d’autres oxydes de métaux de transition 5d ou même

au sein des isolants topologiques.

* * *Interplay of Spin-Orbit Coupling and Electronic Coulomb Interactions


Abstract

In this thesis, we were interested in the interplay between the spin-orbit coupling and electronic correlations in condensed

matter physics. The spin-orbit interaction has indeed been found to play a significant role in the properties of a growing

variety of materials, such as the topological band insulators or the iridium-based 5d-transition metal oxides. Particularly,

strontium iridate (Sr2IrO4) was recently described as a “spin-orbit driven Mott insulator”: according to this picture, the

cooperative interaction between electronic Coulomb interactions and the spin-orbit coupling (ζSO ≈ 0.4 eV) can explain

the insulating state of the compound.

We have studied the paramagnetic insulating phase of this material within LDA+DMFT, a method which combines

the density functional theory in the local density approximation (LDA) with dynamical mean-field theory (DMFT).

Sr2IrO4 was found to be a Mott insulator for a reasonable value of the electronic correlations (U = 1.4 eV) once both

the spin-orbit coupling and the lattice distortions were taken into account. Moreover, our results highlight the respective

roles played by theses two features to reach the Mott insulating state and emphasize that only their acting together may

open the Mott gap in such a compound.

In order to perform this study, the spin-orbit interaction was included in LDA+DMFT – more precisely, to define

the Wannier orbitals on which the local impurity problem is based. The interest of such a technical development goes

beyond the present case of Sr2IrO4 since this “LDA+SO+DMFT implementation” could be also used to take into account

the electronic correlations in the description of other 5d-transition metal oxides or even topological band insulators.

CPhT Ecole Polytechnique – Novembre 2010

interplay of spin-orbit coupling and electronic coulomb

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